Vehicle control method and vehicle control apparatus

ABSTRACT

A target resultant force to be applied to a vehicle body is calculated, the magnitude of a critical friction circle of each wheel is estimated, and a critical resultant force is estimated from the estimated magnitude of the critical friction circle. Subsequently, a ratio of the target resultant force to a critical resultant force is set as an effective road friction, and the magnitude of a tire force is set by using the magnitude of the critical friction circle and the effective road friction. The direction of the tire force of each wheel to be controlled is set based on the sum of products, which are calculated for all other wheels, of a distance from the position of the wheel to be controlled to the position of the other wheel in a direction of the resultant force, and the magnitude of the tire force of the other wheel. Cooperative control of steering and braking or steering and driving of each wheel to be controlled is performed based on the magnitude and direction of the tire force which have been set.

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority under 35 USC 119 from JapanesePatent Application Nos. 2003-24177 and 2003-385973, the disclosures ofwhich are incorporated by reference herein.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to a vehicle control method and avehicle control apparatus. Specifically, the present invention relatesto a method and an apparatus for controlling a vehicle which can controla tire force, that is, a force generated between a tire and a roadsurface, by independently controlling each steering angle and brakingforce of four wheels, and a vehicle which can control a tire force byindependently controlling steering angles and braking forces of frontwheels and those of rear wheels.

[0004] 2. Description of the Related Art

[0005] A technique for separately controlling four wheels and steeringangles thereof has been known, as disclosed in Japanese PatentApplication Laid-Open (JP-A) No. 2001-322557. According to this relatedart, when a vehicle turns, the vehicle is steered so that a steeringangle of each wheel becomes 90° with respect to the center of the turnof the vehicle. Further, when a defective wheel is found, steering andbraking are controlled so that the friction force of the defective wheelis decreased. This related art does not disclose cooperation of steeringand braking, or cooperation of steering and driving. For example,regarding the steering angle, a fixed value is outputted as a targetvalue regardless of braking or driving.

[0006] However, in an actual vehicle, critical friction is generatedbetween a wheel and a road surface, and lateral force may be decreasedby increasing braking force. Therefore, cooperation between steering andbraking or between steering and driving is indispensable in order to usethe friction force between the wheel and the road surface as eficientlyas possible. However, the above related art does not mention thecooperation at all, such as adjusting steering according to braking ordriving. As a result, a problem arises in that the force generated bythe wheel cannot be optimized.

SUMMARY OF THE INVENTION

[0007] An object of the present invention is to solve the aforementionedproblem.

[0008] In order to achieve the above object, a first aspect of thepresent invention may comprise: calculating a physical quantity whichrelates to a tire force of each wheel and optimizes an effective roadfriction of each wheel, based on a target resultant force to be appliedto a vehicle body in order to obtain vehicle body motion that a driverdesires, and a constraint including as parameters a magnitude of acritical friction circle of each wheel; calculating, based on thecalculated physical quantity relating to the tire force of each wheel, afirst control variable for controlling at least one of braking force anddriving force of each wheel, or a second control variable forcontrolling the first control variable and a steering angle of eachwheel; and controlling (A) the at least one of braking force and drivingforce of each wheel based on the first control variable, or controlling(A) the at least one of braking force and driving force of each wheeland (B) the steering angle of each wheel, based on the first and secondcontrol variables.

[0009] The constraint may be represented by a formula indicating that noresultant force is generated in a direction orthogonal to the directionof the target resultant force, and a formula indicating that a momentaround the center of gravity of the vehicle is equal to a desiredmoment. Further, the constraint may be represented by formulae, thenumber of which is less than that of wheels, or a linearized formula.

[0010] The target resultant force may be represented by a secondaryperformance function including a magnitude of the critical frictioncircle of each wheel and the physical quantity relating to the tireforce of each wheel. In such a case, a physical quantity relating to thetire force of each wheel, which physical quantity satisfies a firstapproximation formula of the formula defining the constraint andoptimizes the secondary performance function, can be calculated as thephysical quantity which relates to the tire force of each wheel andoptimizes the effective road friction of each wheel.

[0011] Alternatively, a physical quantity relating to the tire force ofeach wheel, which physical quantity satisfies a first approximationformula of the formula defining the constraint and optimizes thesecondary performance function, is calculated as an initial value. Theformula defining the constraint is linearized by using the calculatedinitial value. Then a physical quantity relating to the tire force ofeach wheel, which physical quantity satisfies the linearized formula ofthe constraint and optimizes the secondary performance function, iscalculated as an approximate solution. The physical quantity whichrelates to the tire force of each wheel and optimizes the effective roadfriction of each wheel can be calculated by using the calculatedapproximate solution as the initial value to repeat the linearization ofthe formula defining the constraint and the calculation of theapproximate solution.

[0012] The formula defining the constraint may be linearized by Taylorexpansion around the initial value or the approximate solution.

[0013] The physical quantity relating to the tire force may be adirection of the tire force. The effective road friction of each wheel,the calculated direction of the tire force of each wheel, and themagnitude of the critical friction circle of each wheel can be used tocalculate a slip angle based on a brush model, and the calculated slipangle can be used to calculate the second control variable based on avehicle motion model.

[0014] The magnitude of the critical friction circle of each wheel canbe determined based on an estimate or a virtual value of a coefficientof friction μ of each wheel and load of each wheel.

[0015] The direction of the tire force which optimizes the effectiveroad friction of each wheel may be one of: a direction of the tire forcewhich uniformly minimizes the effective road friction of each wheel; adirection of the tire force which makes the effective road friction ofthe front wheel differ from that of the rear wheel; and a direction ofthe tire force which makes the magnitude of the tire force of each wheelproportional to the load of the wheel.

[0016] When the magnitude of the tire force proportional to the load ofthe wheel cannot be obtained because each wheel has a different p withrespect to a road surface, the magnitude of the critical friction circlemay be used as the magnitude of the tire force for a wheel having asmall μ, and the magnitude of the tire force which minimizes theeffective road friction may be used for a wheel having a large μ.

[0017] The steering angle may be controlled so as to be the same for theright and left wheels. The effective road friction may be represented bythe magnitude of the target resultant force relative to the magnitude ofa critical resultant force obtained from the magnitude of the criticalfriction circle of each wheel.

[0018] The direction of the tire force which is generated by each wheelmay be defined as a value, that is the sum of products, which arecalculated for all other wheels, of a distance from the position of anobject wheel to the position of the other wheel in the direction of theresultant force, and the magnitude of the critical friction circle ofthe other wheel, with the direction of the resultant force acting on thevehicle body as the resultant force of the tire forces of the respectivewheels being used as a reference.

[0019] A second aspect of the present invention may comprise: targetresultant force calculating means for calculating a target resultantforce to be applied to a vehicle body in order to obtain a vehicle bodymotion that a driver desires; critical friction circle estimating meansfor estimating the magnitude of a critical friction circle of eachwheel; tire force calculating means for calculating a physical quantitywhich relates to a tire force of each wheel and optimizes an effectiveroad friction of each wheel, based on the target resultant force and aconstraint including as parameters the magnitude of the criticalfriction circle of each wheel; control variable calculating means forcalculating, based on the calculated physical quantity relating to thetire force of each wheel, a first control variable for controlling atleast one of braking force and driving force of each wheel, or a secondcontrol variable for controlling the first control variable and asteering angle of each wheel; and control means for controlling (A) theat least one of braking force and driving force of each wheel based onthe first control variable, or controlling (A) the at least one ofbraking force and driving force of each wheel and (B) the steering angleof each wheel based on the first and second control variables.

[0020] Further, a third aspect of the present invention may comprise:target resultant force calculating means for calculating a targetresultant force to be applied to a vehicle body in order to obtain avehicle body motion that a driver desires; critical friction circleestimating means for estimating a magnitude of a critical frictioncircle of each wheel; critical resultant force estimating means forestimating a critical resultant force based on the magnitude of thecritical friction circle of each wheel estimated by the criticalfriction circle estimating means; effective road friction setting meansfor setting a ratio of the target resultant force to the criticalresultant force as an effective road friction; magnitude of tire forcesetting means for setting a magnitude of a tire force used at eachwheel, which tire force is obtained by multiplying the magnitude of thecritical friction circle of each wheel by the effective road friction;direction of tire force setting means for setting a direction of thetire force generated by each wheel based on a value, that is a sum ofproducts, which are calculated for all other wheels, of a distance fromthe position of an object wheel to the position of the other wheel in adirection of the resultant force, and the magnitude of the criticalfriction circle of the other wheel, with the direction of the resultantforce acting on the vehicle body as the resultant force generated by thetire force of each wheel being used as a reference; and control meansfor controlling a steering angle of each wheel and at least one ofbraking force and driving force of each wheel based on the magnitude anddirection of the tire force which have been set.

BRIEF DESCRIPTION OF THE DRAWINGS

[0021]FIG. 1 is a schematic view of a vehicle motion model.

[0022]FIG. 2 is a schematic view of a coordinate system corresponding toa resultant force generated in the vehicle motion model in FIG. 1.

[0023]FIG. 3 is a diagram of a critical friction circle.

[0024]FIG. 4 is a block diagram of a first specific structure accordingto an embodiment of the present invention.

[0025]FIG. 5 is a diagram showing the results of simulation, indicatinga comparison of vehicle accelerations when, in the first specificstructure, the brake is applied to a vehicle which advances straightsuch that one side of the vehicle travels on a snowy road surface andthe other side of the vehicle travels on a dry road surface.

[0026]FIG. 6A is a diagram showing calculated directions of tire forcesof respective wheels, and the number of times calculations are repeated,in the case of front-wheel and rear-wheel steering. FIG. 6B is a diagramshowing calculated directions of tire forces of respective wheels, andthe number of times calculations are repeated, in the case of four-wheelindependent steering.

[0027]FIGS. 7A and 7B are diagrams showing vectors of the tire forces ofthe respective wheels and the steering angles of the respective wheelsafter convergence.

[0028]FIG. 8 is a diagram showing tire force characteristics.

[0029]FIG. 9 is a diagram showing the results of simulation, indicatinga comparison of critical vehicle accelerations when the brake is appliedto a vehicle which is turning such that one side of the vehicle travelson a wet road surface and the other side of the vehicle travels on a dryroad surface.

[0030]FIG. 10A is a diagram showing calculated directions of tire forcesof respective wheels, and the number of times calculations are repeated,in the case of front-wheel and rear-wheel steering. FIG. 10B is adiagram showing calculated directions of tire forces of respectivewheels, and the number of times calculations are repeated, in the caseof four-wheel independent steering.

[0031]FIGS. 11A and 11B are diagrams showing vectors of the tire forcesof the respective wheels and the steering angles of the respectivewheels after convergence.

[0032]FIG. 12 is a diagram showing comparison between tirecharacteristics derived based on a brush model, which theoreticallydescribes the tire force, and actual tire characteristics.

[0033]FIG. 13 is a diagram showing vectors of the tire forces ofrespective wheels and the steering angles of the respective wheels afterconvergence, when the brake is applied to a vehicle which is turning ona road surface of uniform μ.

[0034]FIG. 14 is a block diagram showing a second specific structureaccording to the embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0035] A preferred embodiment of the present invention will be describedin detail below with reference to the drawings. First, a principle ofcooperative control of steering and braking in a vehicle which enablesindependent steering and braking of four wheels, and a principle ofcooperative control of steering and driving in a vehicle which enablesindependent steering and driving of four wheels will be described.

[0036] In a motion model of a four-wheel-drive vehicle shown in FIG. 1,a direction θ in which a force is applied to a vehicle body as aresultant force, which is the sum of forces generated by four wheels, inorder to obtain a vehicle body motion that a driver desires, and themagnitude (radius) F_(i) of a critical friction circle of each wheel(wherein i=1 to 4) are known. (i=1 represents a left front wheel, i=2represents a right front wheel, i=3 represents a left rear wheel, and i=4 represents a right rear wheel.) In this case, the direction of thetire force which maximizes the resultant force, namely, acceleration (ordeceleration) generated by the vehicle body, while securing a desiredyaw moment, is determined for each wheel. The direction of the tireforce of each wheel is represented by an angle q_(i), which is formed bythe direction of the resultant force and the direction of a forcegenerated by a single wheel (i.e., the tire force of each wheel).

[0037] The critical friction circle is a circle representing the limitin which motion performance of the vehicle can be controlled withoutlosing grip of a tire. The magnitude of the critical friction circlerepresents the maximum friction force of a tire generated between thewheel and a road surface, and is determined based on an estimate or avirtual value of μ (friction coefficient) and load of each wheel. Thefriction force of the tire is formed by a force in a direction in whichthe vehicle advances (driving force or braking force) and a frictionforce in a lateral direction (rightward or leftward). The friction forcein either direction becomes 100% of the magnitude of the criticalfriction circle, or corresponds thereto, the friction force in the otherdirection becomes zero. The braking force is applied in a directionopposite to the direction in which the driving force is generated. Therange of the direction force can be represented as a substantialcircular shape when represented as a vector diagram as shown in FIG. 3.For this reason, the range of the friction force is referred to as thecritical friction circle.

[0038] When the motion model of the four-wheel-drive vehicle in FIG. 1is converted into a coordinate shown in FIG. 2 with an X-axis being adirection of the resultant force and a Y-axis being a directionperpendicular to the X-axis, the position of each tire, i.e., (x,y)=(b_(i), a_(i), can be represented by the following formulae (1) to(8). $\begin{matrix}{a_{1} = {{\frac{T_{f}}{2}\cos \quad \theta} - {L_{f}\quad \sin \quad \theta}}} & (1) \\{a_{2} = {{{- \frac{T_{f}}{2}}\cos \quad \theta} - {L_{f}\quad \sin \quad \theta}}} & (2) \\{a_{3} = {{\frac{T_{f}}{2}\cos \quad \theta} + {L_{r}\quad \sin \quad \theta}}} & (3) \\{a_{4} = {{{- \frac{T_{r}}{2}}\cos \quad \theta} + {L_{r}\quad \sin \quad \theta}}} & (4) \\{b_{1} = {{\frac{T_{f}}{2}\sin \quad \theta} + {L_{f}\quad \cos \quad \theta}}} & (5) \\{b_{2} = {{{- \frac{T_{f}}{2}}\sin \quad \theta} + {L_{f}\quad \cos \quad \theta}}} & (6) \\{b_{3} = {{\frac{T_{r}}{2}\sin \quad \theta} - {L_{r}\quad \cos \quad \theta}}} & (7) \\{b_{4} = {{{- \frac{T_{r}}{2}}\sin \quad \theta} - {L_{r}\quad \cos \quad \theta}}} & (8)\end{matrix}$

[0039] In the formulae, T_(f) is an interval between front wheels, Tr isan interval between rear wheels, L_(f) is a distance between the centerof gravity of the vehicle and a middle point of the interval between thefront wheels, and L_(r) is a distance between the center of gravity ofthe vehicle and a middle point of the interval between the rear wheelsas and b_(i) represent a distance from the X-axis and the Y-axis,respectively.

[0040] Further, assuming that M_(z) represents a yaw moment (desiredmoment) to be generated around the center of gravity of the vehicle atthis time, constraints represented by the following formulae (9) and(10) exist for the angle q_(i), which represents the direction of thetire force of each wheel.

F ₁ sin q ₁ +F ₂ sin q ₂ +F ₃ sin q ₃ +F ₄ sin q ₄=0  (9)

−a ₁ F ₁ cos q ₁ −a ₂ F ₂ cos q ₂ −a ₃F₃ cos q ₃ −a ₄ F ₄ cos q ₄ +b ₁ F₁ sin q ₁ +b ₂ F ₂ sin q ₂ +b ₃ F ₃ sin q ₃ +b ₄ F ₄ sin q ₄ =M_(z)  (10)

[0041] The formula (9) represents a constraint indicating that resultantforce is not generated in the Y-axis direction, namely, a directionorthogonal to the direction of the resultant force. The formula (10)represents a constraint indicating that the moment around the center ofgravity of the vehicle is a desired yaw moment M_(z). The number of theformulae representing these constraints is less than the number of thewheels.

[0042] Therefore, the problem of maximizing the resultant force, namely,the problem of maximizing an effective road friction, becomes a problemof determining the angle q_(i) which satisfies the constraints of theformulae (9) and (10) and maximizes the sum (resultant force) J offorces in the X-axis direction represented by the following formula(11).

J=F ₁ cos q ₁ +F ₂ cos q ₂ +F ₃ cos q ₃ +F ₄ cos q ₄  (11)

[0043] This problem can be solved as a nonlinear optimization problem bynumerical calculation of convergence, which will be described later.Alternatively, a solution can be derived from approximation described asfollows.

[0044] First, the formulae (9) and (10) representing the constraints aresubjected to primary approximation to obtain the following formulae (12)and (13).

F ₁ q ₁ +F ₂ q ₂ +F ₃ q ₃ +F ₄ q ₄=0  (12)

b ₁ F ₁ q ₁ +b ₂ F ₂ q ₂ +b ₃ F ₃ b ₃ +b ₄ F ₄ q ₄ =M _(z) +a ₁ F ₁ +a ₂F ₂ +a ₃ F ₃ +a ₄ F ₄  (13)

[0045] The formula (11) is further subjected to secondary approximationto obtain a secondary performance function represented by the followingformula (14). $\begin{matrix}{J = {F_{1} - {\frac{F_{1}}{2}q_{1}^{2}} + F_{2} - {\frac{F_{2}}{2}q_{2}^{2}} + F_{3} - {\frac{F_{3}}{2}q_{3}^{2}} + F_{4} - {\frac{F_{4}}{2}q_{4}^{2}}}} & (14)\end{matrix}$

[0046] Further, the problem of determining the angle q_(i) whichmaximizes the secondary performance function represented by the formula(14) can be replaced with a problem of determining the angle q_(i) whichminimizes the sum (K/2) of negative terms in the formula (14), namely, aproblem of determining the angle q_(i) which minimizes a secondaryperformance function represented by a formula (15).

[0047] Namely, the problem of maximizing the effective road frictionbecomes a problem of determining the angle q_(i) which maximizes thesecondary performance function that is represented by the formula (14)and includes the magnitude of the critical friction circle of each wheeland the direction of the tire force of each wheel, or a problem ofdetermining the angle q_(i) which minimizes the secondary performancefunction that is represented by the following formula (15) and includesthe magnitude of the critical friction circle of each wheel and thedirection of the tire force of each wheel, namely, a problem ofdetermining the angle q_(i) which optimizes the effective road friction.

K=F ₁ q ₁ ² +F ₂ q ₂ ² +F ₃ q ₃ ² +F ₄ q ₄ ²  (15)

[0048] Variables in the formula (15) are changed by using the followingformula (16).

p _(i)={square root}{square root over (F _(i))}q _(i)  (16)

[0049] The formula (15) is represented by the following formula (17).

K=p ₁ ² +p ₂ ² +p ₃ ² +p ₄ ²  (17)

[0050] As a result, the above problem turns out to be replaced with aproblem of determining a value p_(i) of the minimum Euclidean norm whichsatisfies the following formulae (18) and (19).

{square root}{square root over (F ₁)}p ₁+{square root}{square root over(F ₂)}p ₂+{square root}{square root over (F ₃)}p ₃+{square root}{squareroot over (F ₄)}p ₄=0  (18)

b ₁{square root}{square root over (F ₁)}p ₁ +b ₂{square root}{squareroot over (F ₂)}p ₂ +b ₃{square root}{square root over (F ₃)}p ₃ +b₄{square root}{square root over (F ₄)}p ₄ =M _(z) +a ₁ F ₁ +a ₂ F ₂ +a ₃F ₃ +a ₄ F ₄  (19)

[0051] p_(i) can be solved as in the following formula (20).$\begin{matrix}\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack} \cdot \begin{bmatrix}p_{1} \\p_{2} \\p_{3} \\p_{4}\end{bmatrix}}} \\{= {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack} \cdot}} \\{{\left\lbrack \quad \begin{matrix}\sqrt{F_{1}} & \sqrt{F_{2}} & \sqrt{F_{3}} & \sqrt{F_{4}} \\{b_{1}\sqrt{F_{1}}} & {b_{2}\sqrt{F_{2}}} & {b_{3}\sqrt{F_{3}}} & {b_{4}\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack^{+} \cdot}} \\{\begin{bmatrix}0 \\{M_{z} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}}\end{bmatrix}} \\{= {\frac{M_{z} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}}{{b_{1}c_{1}F_{1}} + {b_{2}c_{2}F_{2}} + {b_{3}c_{3}F_{3}} + {b_{4}c_{4}F_{4}}} \cdot \begin{bmatrix}c_{1} \\c_{2} \\c_{3} \\c_{4}\end{bmatrix}}}\end{matrix} & (20)\end{matrix}$

[0052] The formula (20) satisfies the following formulae (21) to (24).

c ₁=(b ₁ −b ₂)F ₂+(b ₁ −b ₃)F ₃+(b ₁ −b ₄)F ₄  (21)

c ₁=(b ₂ −b ₁)F ₁+(b ₂ −b ₃)F ₃+(b ₂ −b ₄)F ₄  (22)

c ₃=(b ₃ −b ₁)F ₁+(b ₃ −b ₂)F ₂+(b ₃ −b ₄)F ₄  (23)

c ₄=(b ₄ −b ₁)F ₁+(b ₄ −b ₂)F ₂+(b ₄ −b ₃)F ₃  (24)

[0053] Further, “diag” in the formula (20) represents a diagonal matrixwith values in the parentheses being diagonal elements. “+” represents apseudoinverse matrix. When A is a long full rank matrix, thepseudoinverse matrix of A can be calculated by the following formula(25).

A ⁺=(A·A ^(T))⁻¹ A ^(T)  (25)

[0054] As will be described later, the angle q_(i) derived from theabove formulae (20) to (24) is directly used as the direction of thetire force of each wheel when the effective road friction γ is 1, inintegrated control of the steering angle and the braking force or thesteering angle and the driving force.

[0055] Further, the formulae (21) to (24) respectively represent the sumof products, which are calculated for all other wheels, of the distancefrom the position of the wheel where the angle q_(i) is determined tothe position of the other wheel in the direction of the resultant forceof the vehicle body (e.g., (b₁−b₂), (b₁−b₃), (b₁−b₄) for a wheel withi=1), and the magnitude of the critical friction circle of the otherwheel. Thus, these formulae represent that the angle q_(i) formed by thedirection of the tire force of each wheel and the direction of theresultant force of the vehicle body is proportional to the sum of theproducts, which are calculated for all other wheels, of the distancefrom the position of the wheel where the angle q_(i) is determined tothe position of the other wheel in the direction of the resultant forceof the vehicle body, and the magnitude of the critical friction circleof the other wheel.

[0056] Furthermore, a_(i) or b_(i) representing the position of eachwheel is a function of the direction θ of the resultant force of thevehicle body. Therefore, the angle q_(i) formed by the direction of thetire force of each wheel and the direction of the resultant force of thevehicle body can be represented as a function of the direction of theresultant force of the vehicle body and the magnitude of the criticalfriction force of each wheel.

[0057] The angle q_(i) derived as described above can also be used as aninitial value for the formulae (9) to (11) for convergence calculationof the nonlinear optimization. Generally, in the nonlinear optimizationproblem, convergence in the convergence calculation tends to accelerateby using a value near the optimal value as an initial value. For thisreason, solutions of the formulae (20) to (24) can be used as theinitial values in the nonlinear optimization problem to increase theefficiency of calculation.

[0058] In the nonlinear optimization problem mentioned herein, Taylorexpansion around approximate solutions derived from the formulae (20) to(24) for the formulae (9) and (10) is carried out to derive solutionstherefrom. Further, Taylor expansion around the solutions for theformulae (9) and (10) is repeated to derive solutions. Consequently,approximate solutions of high accuracy are derived.

[0059] First, the angle q_(i) is derived, as an initial value q_(i0),from the first Taylor expansion, using the formulae (20) to (24), forthe formulae (9) and (10). The following formulae (26) and (27) arederived from Taylor expansion around the initial value q_(i0).

F ₁{sin q ₁₀+(q ₁ −q ₁₀)cos q ₁₀ }+F ₂{sin q ₂₀+(q ₂ −q ₂₀)cos q ₂₀ }+F₃{sin q ₃₀+(q ₃ −q ₃₀)cos q ₃₀ }+F ₄{sin q ₄₀+(q ₄ −q ₄₀)cos q₄₀}=0  (26)

−a ₁ F ₁{cos q ₁₀−(q ₁ −q ₁₀)sin q ₁₀ }−a ₂ F ₂{cos q ₂₀−(q ₂ −q ₂₀)sinq ₂₀ }−a ₃ F ₃{cos q ₃₀−(q ₃ −q ₃₀)sin q ₃₀ }−a ₄ F ₄{cos q ₄₀−(q ₄ −q₄₀)sin q ₄₀ }+b ₁ F ₁{sin q ₁₀+(q ₁ −q ₁₀)cos q ₁₀ }+b ₂ F ₂{sin q ₂₀+(q₂ −q ₂₀)cos q ₂₀ }+b ₃ F ₃{sin q ₃₀+(q ₃ −q ₃₀)cos q ₃₀ }+b ₄ F ₄{sin q₄₀+(q ₄ −q ₄₀)cos q ₄₀}  (27)

[0060] A solution which satisfies these formulae (26) and (27) and alsominimizes the formula (15) is derived from the following formula (28),in which the pseudoinverse matrix described above is used.$\begin{matrix}\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack} \cdot}} \\{{\begin{bmatrix}{\sqrt{F_{1}}\cos \quad q_{10}} & {\sqrt{F_{2}}\cos \quad q_{20}} & {\sqrt{F_{3}}\cos \quad q_{30}} & {\sqrt{F_{4}}\cos \quad q_{40}} \\{\sqrt{F_{1}}\left( {{a_{1}\sin \quad q_{10}} + {b_{1}\cos \quad q_{10}}} \right)} & {\sqrt{F_{2}}\left( {{a_{2}\sin \quad q_{20}} + {b_{2}\cos \quad q_{20}}} \right)} & {\sqrt{F_{3}}\left( {{a_{3}\sin \quad q_{30}} + {b_{3}\cos \quad q_{30}}} \right)} & {\sqrt{F_{4}}\left( {{a_{4}\sin \quad q_{40}} + {b_{4}\cos \quad q_{40}}} \right)}\end{bmatrix}^{+} \cdot}} \\{{\quad\left\lbrack \left. \quad\begin{matrix}{\sum\limits_{i = 1}^{4}{F_{i}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{i0}}} \right)}} \\{M_{z} + {\sum\limits_{i = 1}^{4}{F_{i}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{i0}}} \right\}}}}\end{matrix} \right\rbrack \right.}}\end{matrix} & (28)\end{matrix}$

[0061] Next, solutions which satisfy the formula obtained by the Taylorexpansion around approximate solutions of the formula (28) and minimizethe formula (15) are derived by using the pseudoinverse matrix, asdescribed above. Subsequently, Taylor expansion and derivation ofapproximate solutions are repeated a predetermined number of times toderive an angle q_(i) of high accuracy.

[0062] The accuracy of optimization can be improved by repeatedlyperforming calculations with the formula (28) being used as a recurrenceformula, namely, by using the q_(i), which has been calculated in theprevious step, as q_(i0) in the next step and repeating Taylor expansionand the calculation of q_(i).

[0063] Next, cooperative control of steering and braking and of steeringand driving before a limit will be described. Cooperation of a steeringsystem and a braking system, and cooperation of a steering system and adriving system for improving critical motion performance (i.e., theforce generated by the vehicle body) have been described above. Acooperation method for extending the control rules obtained above to arange before the limit and maximizing grip of each wheel will bedescribed below.

[0064] When the effective road friction of each wheel is expressed as γ,constraints of a force generated by the vehicle body in a lateraldirection and a yaw direction are represented by the following formulae,which are similar to the formulae (9) and (10), respectively

γF ₁ sin q ₁ +γF ₂ sin q ₂ +γF ₃ sin q ₃ +γF ₄ sin q ₄=0

−a ₁ γF ₁ cos q ₁ −a ₂ γF ₂ cos q ₂ −a ₃ γF ₃ cos q ₃ −a ₄ γF ₄ cos q₄+b ₁ γF ₁ sin q ₁ +b ₂ γF ₂ sin q ₂ +b ₃ γF ₃ sin q ₃ +b ₄ γF ₄ sin q ₄=M _(z)

[0065] Namely, the constraints are represented by the following formulae(29) and (30).

F ₁ sin q ₁ +F ₂ sin q ₂ +F ₃ sin q ₃ +F ₄ sin q ₄=0  (29)$\begin{matrix}{{{{- a_{1}}F_{1}\cos \quad q_{1}} - {a_{2}F_{2}\cos \quad q_{2}} - {a_{3}F_{3}\cos \quad q_{3}} - {a_{4}F_{4}\cos \quad q_{4}} + {b_{1}F_{1}\sin \quad q_{1}} + {b_{2}F_{2}\sin \quad q_{2}} + {b_{3}F_{3}\sin \quad q_{3}} + {b_{4}F_{4}\sin \quad q_{4}}} = \frac{M_{z}}{\gamma}} & (30)\end{matrix}$

[0066] Further, the magnitude of the force generated by the vehicle bodyis represented by the following formula as a constraint.

γF ₁ cos q ₁ +γF ₂ cos q ₂ +γF ₃ cos q ₃ +γF ₄ cos q ₄ =F

[0067] Namely, the constraint is represented by the following formula(31). $\begin{matrix}{{{F_{1}\cos \quad q_{1}} + {F_{2}\cos \quad q_{2}} + {F_{3}\cos \quad q_{3}} + {F_{4}\cos \quad q_{4}}} = \frac{F}{\gamma}} & (31)\end{matrix}$

[0068] Therefore, the cooperation method for uniformly maximizing thegrip of each wheel becomes a problem of determining the angle q_(i)which satisfies the formulae (29) to (31) and minimizes the effectiveroad friction γ. Further, when F≠0, the problem can be considered as aproblem of determining the angle q_(i) which satisfies the above formula(29) and the following formula (32) obtained by organizing the formulae(30) and (31), and maximizes the following formula (33). $\begin{matrix}{{{b_{1}F_{1}\sin \quad q_{1}} + {b_{2}F_{2}\sin \quad q_{2}} + {b_{3}F_{3}\sin \quad q_{3}} + {b_{4}F_{4}\sin \quad q_{4}}} = {{\left( {a_{1} + \frac{M_{z}}{F}} \right)F_{1}\cos \quad q_{1}} + {\left( {a_{2} + \frac{M_{z}}{F}} \right)F_{2}\cos \quad q_{2}} + {\left( {a_{3} + \frac{M_{z}}{F}} \right)F_{3}\cos \quad q_{3}} + {\left( {a_{4} + \frac{M_{z}}{F}} \right)F_{4}\cos \quad q_{4}}}} & (32) \\{J = {\frac{F}{\gamma} = {{F_{1}\cos \quad q_{1}} + {F_{2}\cos \quad q_{2}} + {F_{3}\cos \quad q_{3}} + {F_{4}\cos \quad q_{4}}}}} & (33)\end{matrix}$

[0069] When an approximate solution is derived in the same way asdescribed above, the formulae (29) and (32) are represented by thefollowing formulae (34) and (35), respectively, after primaryapproximation.

F ₁ q ₁ +F ₂ q ₂ +F ₃ q ₃ +F ₄ q ₄=0  (34) $\begin{matrix}{{{b_{1}F_{1}q_{1}} + {b_{2}F_{2}q_{2}} + {b_{3}F_{3}q_{3}} + {b_{4}F_{4}q_{4}}} = {{\left( {a_{1} + \frac{M_{z}}{F}} \right)F_{1}} + {\left( {a_{2} + \frac{M_{z}}{F}} \right)F_{2}} + {\left( {a_{3} + \frac{M_{z}}{F}} \right)F_{3}} + {\left( {a_{4} + \frac{M_{z}}{F}} \right)F_{4}}}} & (35)\end{matrix}$

[0070] Further, the formula (33) corresponds with the secondaryperformance function of the formula (14) by secondary approximation. Forthis reason, as described above, variables are changed by using theformula (16), and the problem is replaced with a problem of determininga value p_(i) of the minimum Euclidean norm which satisfies thefollowing formulae (36) and (37).

{square root}{square root over (F ₁)}p ₁+{square root}{square root over(F ₂)}p ₂+{square root}{square root over (F ₃)}p ₃+{square root}{squareroot over (F ₄)}p ₄=0  (36) $\begin{matrix}{{{b_{1}\sqrt{F_{1}}p_{1}} + {b_{2}\sqrt{F_{2}}p_{2}} + {b_{3}\sqrt{F_{3}}p_{3}} + {b_{4}\sqrt{F_{4}}p_{4}}} = {{\left( {a_{1} + \frac{M_{z}}{F}} \right)F_{1}} + {\left( {a_{2} + \frac{M_{z}}{F}} \right)F_{2}} + {\left( {a_{3} + \frac{M_{z}}{F}} \right)F_{3}} + {\left( {a_{4} + \frac{M_{z}}{F}} \right)F_{4}}}} & (37)\end{matrix}$

[0071] The value p_(i) can be solved by the following formula (38).$\begin{matrix}\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack} \cdot \begin{bmatrix}p_{1} \\p_{2} \\p_{3} \\p_{4}\end{bmatrix}}} \\{= {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack} \cdot}} \\{{\left\lbrack \quad \begin{matrix}\sqrt{F_{1}} & \sqrt{F_{2}} & \sqrt{F_{3}} & \sqrt{F_{4}} \\{b_{1}\sqrt{F_{1}}} & {b_{2}\sqrt{F_{2}}} & {b_{3}\sqrt{F_{3}}} & {b_{4}\sqrt{F_{4}}}\end{matrix}\quad \right\rbrack^{+} \cdot}} \\{{\quad\left\lbrack {{\frac{M_{z}}{F}\left( {F_{1} + F_{2} + F_{3} + F_{4}} \right)} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \right\rbrack}} \\{= {\frac{\begin{matrix}{{\frac{M_{z}}{F}\left( {F_{1} + F_{2} + F_{3} + F_{4}} \right)} + {a_{1}F_{1}} +} \\{{a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}}\end{matrix}}{{b_{1}c_{1}F_{1}} + {b_{2}c_{2}F_{2}} + {b_{3}c_{3}F_{3}} + {b_{4}c_{4}F_{4}}} \cdot \begin{bmatrix}c_{1} \\c_{2} \\c_{3} \\c_{4}\end{bmatrix}}}\end{matrix} & (38)\end{matrix}$

[0072] The formula (38) satisfies the following formulae (39) to (42).

c ₁=(b ₁ −b ₂)F ₂+(b ₁ −b ₃)F ₃+(b ₁ −b ₄)F ₄  (39)

c ₂=(b ₂ −b ₁)F ₁+(b ₂ −b ₃)F ₃+(b ₂ −b ₄)F ₄  (40)

c ₃=(b ₃ −b ₁)F ₁+(b ₃ −b ₂)F ₂+(b ₃ −b ₄)F ₄  (41)

c ₄=(b ₄ −b ₁)F ₁+(b ₄ −b ₂)F ₂+(b ₄ −b ₃)F ₃  (42)

[0073] As described above, “diag” represents a diagonal matrix, and “+”represents a pseudoinverse matrix. The angle q_(i) derived from theformulae (38) to (42) may be directly used in integrated control ofsteering and braking or steering and driving, as the direction of thetire force of each wheel, or may be used as an initial value for theformulae (29) to (31) for convergence calculation of the nonlinearoptimization. In the same way as described above, a recurrence formulafor determining an approximate solution of high accuracy by repeatedcalculation based on the Taylor expansion is given as the followingformula (43). $\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{diag}\left\lbrack {\begin{matrix}\frac{1}{\sqrt{F_{1\quad}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3\quad}}} & \left. \frac{1}{\sqrt{F_{4}}} \right\rbrack\end{matrix} \cdot {\begin{bmatrix}{\sqrt{F_{1}}\cos \quad q_{10}} & {\sqrt{F_{2}}\cos \quad q_{20}} & {\sqrt{F_{3}}\cos \quad q_{30}} & {\sqrt{F_{4}}\cos \quad q_{40}} \\{\sqrt{F_{1}}\left( {{a_{1}\sin \quad q_{10}} + {b_{1}\quad \cos \quad q_{10}}} \right)} & {\sqrt{F_{2}}\left( {{a_{2}\sin \quad q_{20}} + {b_{2}\quad \cos \quad q_{20}}} \right)} & {\sqrt{F_{3}}\left( {{a_{3}\sin \quad q_{30}} + {b_{3}\quad \cos \quad q_{30}}} \right)} & {\sqrt{F_{4}}\left( {{a_{4}\sin \quad q_{40}} + {b_{4}\quad \cos \quad q_{40}}} \right)}\end{bmatrix}^{+}.{\quad\begin{bmatrix}{\sum\limits_{i = 1}^{4}\quad {F_{i}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{{i0}\quad}}} \right)}} \\{\sum\limits_{i = 1}^{4}\quad {F_{i}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}} + \frac{M_{z}}{F}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{io}}} \right\}}}\end{bmatrix}}}} \right.}} & (43)\end{matrix}$

[0074] The effective road friction γ is calculated based on the formula(44) in which the angle q_(i) thus derived is used. Namely, theeffective road friction γ is represented as a ratio of a targetresultant force (i.e., force of the vehicle body) F to a criticalresultant force obtained from the magnitude of the critical frictioncircle of each wheel (i.e., resultant force in the direction of thetarget resultant force, which resultant force is the sum of criticalforces obtained from the magnitude of the critical friction circle ofeach wheel). $\begin{matrix}{\gamma = \frac{F}{{F_{1}\cos \quad q_{1}} + {F_{2}\cos \quad q_{2}} + {F_{3}\cos \quad q_{3}} + {F_{4\quad}\cos \quad q_{4}}}} & (44)\end{matrix}$

[0075] Moreover, braking or driving force of each wheel is derived fromthe following formula (45) by using the effective road friction γ, andthe magnitude F_(i) of the critical friction circle of each wheel, andthe direction (q_(i)+θ) of the tire force of each wheel. γF_(i)represents the magnitude of the tire force.

F _(xi) =γF _(i) cos(q _(i)+θ)  (45)

[0076] Further, lateral force applied to each wheel is derived from thefollowing formula (46).

F _(yi) =γF _(i) sin(q _(i)+θ)  (46)

[0077] The steering angle of each wheel is calculated based on, forexample, a brush model and a vehicle motion model. The brush model is amodel which describes a characteristic of the tire force based on atheoretical formula. Assuming that the tire force is generated inaccordance with the brush model, a slip angle β_(i) can be determined bythe following formula (47), using the magnitude of the critical frictioncircle of each wheel, the effective road friction γ, and the directionof the tire force of each wheel (q_(i)+θ). $\begin{matrix}{\beta_{i} = {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{i}}{\sin \left( {q_{i} + \theta} \right)}}{1 - {k_{i}\cos \quad \left( {q_{i} + \theta} \right)}}} \right)}} & (47)\end{matrix}$

[0078] The slip angle β_(i) satisfies the following formula (48).$\begin{matrix}{k_{i} = {\frac{3F_{i}}{K_{s}}\left( {1 - \left( {1 - \gamma} \right)^{\frac{1}{3}}} \right)}} & (48)\end{matrix}$

[0079] In the above formulae (47) and (48), K_(s) represents drivingstiffness, and K_(β) represents cornering stiffness.

[0080] Further, a steering angle δ_(i) of each wheel is calculated fromthe slip angle based on the vehicle motion model. Namely, the steeringangle can be calculated as in the following formulae (49) to (52) byusing yaw angle speed r₀ and a vehicle body slip angle β₀, which arecalculated as target vehicle motion state variables from vehicle speedv, the steering angle, accelerator stroke, braking effort, and the like.β₁ to β₄ represent the slip angles of the respective wheels in theformula (47). $\begin{matrix}{\delta_{1} = {\beta_{0} + {\frac{L_{f}}{v}r_{0}} - \beta_{1}}} & (49) \\{\delta_{2} = {\beta_{0} + {\frac{L_{f}}{v}r_{0}} - \beta_{2}}} & (50) \\{\delta_{3} = {\beta_{0} - {\frac{L_{f}}{v}r_{0}} - \beta_{3}}} & (51) \\{\delta_{4} = {\beta_{0} - {\frac{L_{f}}{v}r_{0}} - \beta_{4}}} & (52)\end{matrix}$

[0081] When cooperation of steering control and braking and drivingcontrol is carried out, braking force and driving force are controlledbased on first control variables, which are the braking force anddriving force determined by the above formula (45). The steering angle,namely, the direction of the tire force, is controlled based on secondcontrol variables, which are the steering angles determined in theformulae (49) to (52). Alternatively, either the braking force or thedriving force may be controlled while the direction of the tire force iscontrolled.

[0082] When the cooperation of the steering control and the braking anddriving control is carried out based on this type of control, theeffective road friction γ of each wheel can be uniformly minimized allthe time, and motion performance allowing the greatest robust againstdisturbance such as a road surface or cross wind can be obtained.

[0083] Further, when the resultant force to be applied to the vehiclebody is maximized, the braking force and the driving force of each wheelcan be determined by the formula (45) when the effective road friction γof each wheel is 1, and the steering angles of the respective wheels canbe determined by the formulae (49) to (52) when the effective roadfriction γ of each wheel in the formula (48) is 1.

[0084] The braking force, the driving force and the steering angle ofeach wheel determined above are used as the control variables to controlthe cooperation of the driving force and the steering angle of thevehicle or the cooperation of the braking force and the steering angle.

[0085] Only the braking force and the driving force may be calculated tocontrol the driving force, the braking force, or both the braking forceand the driving force. Namely, only the magnitude of the tire force maybe controlled without controlling the steering angle.

[0086] Next, a case will be described in which the cooperative controlis applied to a normal four-wheel vehicle and carried out by using thesame steering angle for the right and left wheels. In the case of theconventional four-wheel vehicle having the same steering angle for theright and left wheels, constraints represented by the following formulae(53) and (54) and indicating that the right and left wheels have thesame slip angle are further added. $\begin{matrix}{{\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{1}}{\sin \left( {q_{1} + \theta} \right)}}{1 - {k_{1}\cos \quad \left( {q_{1} + \theta} \right)}}} \right)} = {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{2}}{\sin \left( {q_{2} + \theta} \right)}}{1 - {k_{2}\cos \quad \left( {q_{2} + \theta} \right)}}} \right)}} & (53) \\{{\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{3}}{\sin \left( {q_{3} + \theta} \right)}}{1 - {k_{3}\cos \quad \left( {q_{3} + \theta} \right)}}} \right)} = {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{4}}{\sin \left( {q_{4} + \theta} \right)}}{1 - {k_{4}\cos \quad \left( {q_{4} + \theta} \right)}}} \right)}} & (54)\end{matrix}$

[0087] These constraints can be organized into the following formulae(55) and (56).

k ₂ sin(q ₂+θ)−k ₁ sin(q ₁+θ)−k ₁ k ₂ sin(q ₂ −q ₁)=0  (55)

k ₄ sin(q ₄+θ)−k ₃ sin(q ₃+θ)−k ₃ k ₄ sin(q ₄ −q ₃)=0  (56)

[0088] After primary approximation, these constraints are represented bythe following formulae (57) and (58).

−k ₁(cos θ−k ₂)q ₁ +k ₂(cos θ−k ₁)q ₂=(k ₁ −k ₂)sin θ  (57)

−k ₃(cos θ−k ₄)q ₃ +k ₄(cos θ−k ₃)q ₄=(k ₃ −k ₄)sin θ  (58)

[0089] As described above, constraints such as the following formulae(59) and (60) are used when the effective road friction γ is 1, namely,when the force of the vehicle body is maximized.

F ₁ sin q ₁ +F ₂ sin q ₂ +F ₃ sin q ₃ −a ₄ F ₄ cos q ₄=0  (59)

−a ₁ F ₁ cos q ₁ −a ₂ F ₂ cos q ₂ −a ₃ F ₃ cos q ₃ −a ₄ F ₄ cos q ₄ +b ₁F ₁ sin q ₁ +b ₂ F ₂ sin q ₂ +b ₃ F ₃ sin q ₃ +b ₄ F ₄ sin q ₄ =M_(z)  (60)

[0090] Therefore, an approximate solution is derived from the followingformula (61). $\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {\quad{\quad{\left\lbrack {\begin{matrix}\begin{matrix}F_{1} \\{{b_{1}F_{1}}\quad} \\{- {k_{1}\left( {{\cos \quad \theta} - k_{2}} \right)}} \\0\end{matrix} & {{\begin{matrix}F_{2} \\{{b_{2}F_{2}}\quad} \\{k_{2}\left( {{\cos \quad \theta} - k_{1}} \right)} \\0\end{matrix}\begin{matrix}F_{3} \\{{b_{3}F_{3}}\quad} \\0 \\{- {k_{3}\left( {{\cos \quad \theta} - k_{4}} \right)}}\end{matrix}}\quad}\end{matrix}\begin{matrix}F_{4} \\{{b_{4}F_{4}}\quad} \\0 \\{k_{4}\left( {{\cos \quad \theta} - k_{3}} \right)}\end{matrix}} \right\rbrack^{- 1} \cdot {\quad\begin{bmatrix}0 \\{M_{z} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta} \\{\left( {k_{3} - k_{4}} \right)\sin \quad \theta}\end{bmatrix}}}}}} & (61)\end{matrix}$

[0091] Further, when the force F of the vehicle body is applied with theeffective road friction γ being 1 or less, the formula (32) is usedinstead of the formula (60). Furthermore, the effective road friction γis determined by the following formula.$\gamma = \frac{F}{F_{1} + F_{2} + F_{3} + F_{4}}$

[0092] An approximate solution is derived from the following formula(62) by using the effective road friction γ determined above.$\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {\quad{\quad{\left\lbrack {\begin{matrix}\begin{matrix}F_{1} \\{{b_{1}F_{1}}\quad} \\{- {k_{1}\left( {{\cos \quad \theta} - k_{2}} \right)}} \\0\end{matrix} & {{\begin{matrix}F_{2} \\{{b_{2}F_{2}}\quad} \\{k_{2}\left( {{\cos \quad \theta} - k_{1}} \right)} \\0\end{matrix}\begin{matrix}F_{3} \\{{b_{3}F_{3}}\quad} \\0 \\{- {k_{3}\left( {{\cos \quad \theta} - k_{4}} \right)}}\end{matrix}}\quad}\end{matrix}\begin{matrix}F_{4} \\{{b_{4}F_{4}}\quad} \\0 \\{k_{4}\left( {{\cos \quad \theta} - k_{3}} \right)}\end{matrix}} \right\rbrack^{- 1} \cdot {\quad\begin{bmatrix}0 \\{{\frac{M_{z}}{F}\left( {F_{1} + F_{2} + F_{3} + F_{4}} \right)} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta}\end{bmatrix}}}}}} & (62)\end{matrix}$

[0093] The braking force, the driving force and the steering angle atthis time are calculated in the formulae (45) and (47) to (52). However,the same value is obtained in the formulae (46) to (49) for the steeringangles of the right and left wheels. The solutions of the formulae (61)and (62) are derived by primary approximation. As described above, thesesolutions may be used as the initial values for the correspondingnonlinear equation for numerical calculation, and the control based on asolution thus obtained may be carried out. When the repeatedcalculations using the above-described Taylor expansion are applied tothe formula (61), an approximate solution is represented by thefollowing formula (63). $\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {\quad{\quad\left\lbrack {\begin{matrix}\begin{matrix}{F_{1}\cos \quad q_{10}} \\{{a_{1}F_{1}\sin \quad q_{10}} + {b_{1}F_{1}\quad \cos \quad q_{10}} -} \\{{- k_{1}}\left\{ {{\cos \left( {q_{10} + \theta} \right)} - {k_{2}{\cos \left( {q_{20} - q_{10}} \right)}}} \right\}} \\0\end{matrix} & \begin{matrix}{F_{2}\cos \quad q_{20}} \\{{a_{2}F_{2}\sin \quad q_{20}} + {b_{2}F_{2}\quad \cos \quad q_{20}}} \\{k_{2}\left\{ {{\cos \left( {q_{20} + \theta} \right)} - {k_{1}{\cos \left( {q_{20} - q_{10}} \right)}}} \right\}} \\0\end{matrix}\end{matrix}{\begin{matrix}\begin{matrix}{F_{3}\cos \quad q_{30}} \\{{a_{3}F_{3}\sin \quad q_{30}} + {b_{3}F_{3}\quad \cos \quad q_{30}} -} \\0 \\{k_{3}\left\{ {{\cos \left( {q_{30} + \theta} \right)} - {k_{4}{\cos \left( {q_{40} - q_{30}} \right)}}} \right\}}\end{matrix} & \left. \begin{matrix}{F_{4}\cos \quad q_{40}} \\{{a_{4}F_{4}\sin \quad q_{40}} + {b_{4}F_{4}\quad \cos \quad q_{40}}} \\0 \\{k_{4}\left\{ {{\cos \left( {q_{40} + \theta} \right)} - {k_{3}{\cos \left( {q_{40} - q_{30}} \right)}}} \right\}}\end{matrix} \right\rbrack^{- 1}\end{matrix} \cdot {\quad\begin{bmatrix}{\sum\limits_{i = 1}^{4}\quad {F_{i}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{{i0}\quad}}} \right)}} \\{M_{z} + {\sum\limits_{i = 1}^{4}\quad {F_{i}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{io}}} \right\}}}} \\d_{1} \\d_{2}\end{bmatrix}}}} \right.}}} & (63)\end{matrix}$

[0094] The formula (63) satisfies the following formula.d₁ = k₁{sin (q₁₀ + θ) − q₁₀cos (q₁₀ + θ)} − k₂{sin (q₂₀ + θ) − q₂₀cos (q₂₀ + θ)} + k₁k₂{sin (q₂₀ − q₁₀) − (q₂₀ − q₁₀)cos (q₂₀ − q₁₀)}d₂ = k₃{sin (q₃₀ + θ) − q₃₀cos (q₃₀ + θ)} − k₄{sin (q₄₀ + θ) − q₄₀cos (q₄₀ + θ)} + k₃k₄{sin (q₄₀ − q₃₀) − (q₄₀ − q₃₀)cos (q₄₀ − q₃₀)}

[0095] q_(i0): value obtained in the previous step

[0096] Further, when the repeated calculations using the Taylorexpansion are applied to the formula (62), an approximate solution isrepresented by the following formula (64). $\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {\quad{\quad\left\lbrack {\begin{matrix}\begin{matrix}{F_{1}\cos \quad q_{10}} \\{{a_{1}F_{1}\sin \quad q_{10}} + {b_{1}F_{1}\quad \cos \quad q_{10}} -} \\{k_{1}\left\{ {{\cos \left( {q_{10} + \theta} \right)} - {k_{2}{\cos \left( {q_{20} - q_{10}} \right)}}} \right\}} \\0\end{matrix} & \begin{matrix}{F_{2}\cos \quad q_{20}} \\{{a_{2}F_{2}\sin \quad q_{20}} + {b_{2}F_{2}\quad \cos \quad q_{20}}} \\{k_{2}\left\{ {{\cos \left( {q_{20} + \theta} \right)} - {k_{1}{\cos \left( {q_{20} - q_{10}} \right)}}} \right\}} \\0\end{matrix}\end{matrix}{\begin{matrix}\begin{matrix}{F_{3}\cos \quad q_{30}} \\{{a_{3}F_{3}\sin \quad q_{30}} + {b_{3}F_{3}\quad \cos \quad q_{30}}} \\0 \\{k_{3}\left\{ {{\cos \left( {q_{30} + \theta} \right)} - {k_{4}{\cos \left( {q_{40} - q_{30}} \right)}}} \right\}}\end{matrix} & \left. \begin{matrix}{F_{4}\cos \quad q_{40}} \\{{a_{4}F_{4}\sin \quad q_{40}} + {b_{4}F_{4}\quad \cos \quad q_{40}}} \\0 \\{k_{4}\left\{ {{\cos \left( {q_{40} + \theta} \right)} - {k_{3}{\cos \left( {q_{40} - q_{30}} \right)}}} \right\}}\end{matrix} \right\rbrack^{- 1}\end{matrix} \cdot {\quad\begin{bmatrix}{\sum\limits_{i = 1}^{4}\quad {F_{i}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{{i0}\quad}}} \right)}} \\{\sum\limits_{i = 1}^{4}\quad {F_{i}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}} + \frac{M_{z}}{F}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{io}}} \right\}}} \\d_{1} \\d_{2}\end{bmatrix}}}} \right.}}} & (64)\end{matrix}$

[0097] Next, the first specific structure of the present embodimentusing the above principle will be described based on FIG. 4. As shown inthe drawing, the present embodiment includes a target resultant forcecalculating unit 18 for calculating the magnitude and direction of atarget resultant force; a critical friction circle estimating unit 20for estimating the magnitude (radius) of a critical friction circle ofeach wheel; a direction of tire force and effective road frictioncalculating unit 25 for calculating the direction of force generated byeach wheel and the effective road friction based on the magnitude anddirection of the target resultant force and the magnitude of thecritical friction force of each wheel; a tire force setting unit 28 forsetting the magnitude of the tire force generated by each wheel; and acontrol unit 30 connected to a cooperative braking and drivingapparatus.

[0098] The target resultant force calculating unit 18 calculates, fromthe steering angle, the vehicle speed, the accelerator stroke, thebraking effort, and the like, the magnitude and direction of a resultantforce and a yaw moment to be applied to the vehicle body in order toobtain a vehicle body motion that the driver desires. The magnitude anddirection of the resultant force and the yaw moment can be determinedby, for example, calculating the magnitude and direction of theresultant force and the yaw moment which are necessary to approximate tozero deviations from a yaw angle speed, which is a target vehicle motionstate variable set in accordance with the operation of the driver, andfrom a measured value (or an estimate) of the slip angle of the vehiclebody.

[0099] The critical friction circle estimating unit 20 estimates themagnitude of the critical friction circle of each wheel based on SAT(self-aligning torque) or a wheel speed.

[0100] The direction of the tire force and effective road frictioncalculating unit 25 calculates the direction of the tire force and theeffective road friction of each wheel based on the above constraintsincluding as parameters the magnitude and direction of the targetresultant force and the magnitude of the critical friction circle ofeach wheel. In this case, the direction of the tire force may becalculated so that the effective road friction of each wheel isuniformly minimized, or may be calculated so that the magnitude of thetire force of each wheel is minimized in accordance with a loaddistribution ratio.

[0101] When the magnitude of the tire force proportional to a wheel loadcannot be calculated because each wheel has a different μ with respectto the road surface, the magnitude of the critical friction circle maybe used as the magnitude of the tire force of the wheel having a smallμ, and the magnitude of the tire force obtained with the smallesteffective road friction may be used for the wheel having a large μ.

[0102] As described above, the direction of the tire force of each wheelcan be determined based on linear algebraic equations, the number ofwhich is less than the number of wheels. Further, when the effectiveroad friction of each wheel is uniformly minimized, as described above,the direction of the tire force can be determined based on the sum ofproducts, which are calculated for all other wheels, of the distancefrom the position of an object wheel, the direction of the force ofwhich is to be determined, to other wheel, which distance is in adirection in which a resultant force is applied to the vehicle body asthe resultant force of the tire forces, and the magnitude of thecritical friction circle of the other wheel.

[0103] The tire force setting unit 28 calculates the steering angle, thebraking force and the driving force of each wheel based on the directionof the tire force of each wheel for optimizing the effective roadfriction.

[0104] The control unit 30 controls a steering actuator, a brakingactuator and a driving actuator so that the steering angle, the brakingforce, and the driving force calculated by the tire force setting unit28 are obtained.

[0105]FIG. 5 is a diagram showing the results of calculation, bysimulation, of the comparison of vehicle accelerations [G] when, in thefirst specific structure described above, the brake is applied to thevehicle which advances straight such that one side of the vehicletravels on a snowy road surface having a μ of 0.3 and the other side ofthe vehicle travels on a dry road surface having a μ of 1.0. In thiscase, control based on the formulae (28) and (63) is carried out infour-wheel independent steering and front-wheel and rear-wheel steering,respectively, and it can be understood that braking force is increasedby 8% in the four-wheel independent steering.

[0106]FIGS. 6A and 6B are diagrams showing the number of calculationsrepeated of the direction of the tire force, which calculations arebased on the formulae (28) and (63), respectively. FIGS. 7A and 7B arediagrams each showing a vector of the tire force and a steering angle ofeach wheel after convergence.

[0107] It can be understood from FIGS. 6A and 6B that the direction ofthe force generated by each wheel converges in a second or thirdcalculation. Further, it can be understood from FIG. 7B that, in thecase of the four-wheel independent steering, a lateral force canceling amoment is generated at a wheel having a high μ, which wheel hasrelatively large friction allowance.

[0108] The front-wheel and rear-wheel steering shown in FIG. 7A adopts acontrol rule which uses an edge of the friction circle (k_(a) in a tirecharacteristics shown in FIG. 8). However, considering the use of a sliparea (k_(b) in the tire characteristics shown in FIG. 8) which isfurther than the edge, the direction of the tire force can be changedwithout changing the steering angle.

[0109] In this case, the direction of the tire force of each wheel,which is the solution for the four-wheel independent steering, can berealized by a front-wheel and rear-wheel steering vehicle. Theacceleration generated by the vehicle body which is equivalent to thatof the four-wheel independent steering can be obtained by the followinglogic.

[0110] First, a slip angle and a steering angle are determined based onthe calculated value of the direction of the tire force of a wheelhaving a high μ (a control rule of the four-wheel independent steering).The wheel of high μ outputs a braking force and a driving force whichare the same as those in the four-wheel independent steering.

[0111] Next, a slip ratio for making the slip angles of the wheels ofhigh μ correspond to each other and obtaining the direction of the tireforce calculated from the control rule of the four-wheel independentsteering is determined as follows. When a low μ and a high μ arerepresented by subscripts i and j, respectively, a condition in whichthe slip angles of the right and left wheels correspond to each other isrepresented by the following formula: $\begin{matrix}{{{provided}\quad {that}\quad {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{i}}{\sin \left( {q_{i} + \theta} \right)}}{1 - {k_{i}\cos \quad \left( {q_{i} + \theta} \right)}}} \right)}} = {{{\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{j}}{\sin \left( {q_{j} + \theta} \right)}}{1 - {k_{j}\cos \quad \left( {q_{j} + \theta} \right)}}} \right)}\quad k_{j}} = {\frac{3F_{j}}{K_{s}}.}}} & (65)\end{matrix}$

[0112] In the above formula (65), q_(i) and q_(j) are solutions(directions of the tire force) derived by assuming the four-wheelindependent steering. These solutions are used as follows to solvek_(i). $\begin{matrix}{k_{i} = \frac{3F_{j}{\sin \left( {q_{j} + \theta} \right)}}{{K_{s}{\sin \left( {q_{i} + \theta} \right)}} + {3F_{j}{\sin \left( {q_{j} - q_{i}} \right)}}}} & (66)\end{matrix}$

[0113] The slip ratio (in a longitudinal direction of the vehicle) iscalculated as follows. $\begin{matrix}{k_{xi} = {{k_{i}\quad {\cos \left( {q_{i} + \theta} \right)}} = \frac{3F_{j}{\sin \left( {q_{j} + \theta} \right)}{\cos \left( {q_{i} + \theta} \right)}}{{K_{s}{\sin \left( {q_{i} + \theta} \right)}} + {3F_{j}{\sin \left( {q_{j} - q_{i}} \right)}}}}} & (67)\end{matrix}$

[0114] Further, the braking force or the driving force is given by thefollowing formula.

F _(xi) =F _(i) cos(q _(i)+θ)  (68)

[0115]FIG. 9 is a diagram showing the results of calculating, bysimulation, the comparison of critical accelerations [G] (the directionsof the resultant force) during such turning and braking that theresultant force is generated in a direction of θ=120° on a wet roadsurface having a μ of 0.8 for outer wheels and a dry road surface havinga μ of 1.0 for inner wheels.

[0116] “FRONT-WHEEL AND REAR-WHEEL STEERING” indicates an accelerationof the vehicle when the steering angles of the front and rear wheels andthe braking forces of the respective four wheels are determined based onthe formula (63) in the vehicle structured such that the right wheel andthe left wheel are steered at the same angle. Further, “FOUR-WHEELINDEPENDENT STEERING” indicates an acceleration of a vehicle when thedirection of the force generated by each wheel is determined based onthe formula (28). It can be seen that the acceleration generated by thevehicle body under these conditions is increased by 5% in the four-wheelindependent steering.

[0117]FIGS. 10A and 10B are diagrams showing the number of calculationsrepeated of the direction of the tire force of each wheel, whichcalculations are based on the formulae (28) and (63), respectively.FIGS. 11A and 11B are diagrams each showing a vector of the tire forceand an steering angle of each wheel after convergence. It can be seenthat, in the case of the four-wheel independent steering, the steeringangles of the outer wheels having a small μ are smaller than those ofthe inner wheels. Cornering power (CP), dependency of lateral force onthe load, a change in susgeometry, and the like are not considered inthis simulation.

[0118] In this simulation as well, the front-wheel and rear-wheelsteering shown in FIG. 10A adopts the control rule which uses the edgeof the friction circle (k_(a) in the tire characteristic in FIG. 8).However, considering the use of the slip area further than the edge(k_(b) in the tire characteristics in FIG. 8), the direction of the tireforce can be changed without changing the steering angle. Therefore, asin the case of the vehicle advancing straight such that the one side andthe other side of the vehicle travel on different types of road surface,the front-wheel and rear-wheel steering vehicle, in which the right andleft wheels are controlled so as to have the same steering angle, canalso obtain a vehicle acceleration which is equivalent to that of thefour-wheel-drive vehicle. As described above, in a critical travelingstate (wherein the effective road friction =1), the front-wheel andrear-wheel steering vehicle in which the right and left wheels have thesame steering angle can also obtain the vehicle acceleration which isequivalent to that of the four-wheel independent steering by using theslip area further than the edge of the friction circle. This is theresult of derivation based on the brush model which theoreticallydescribes the tire force. However, in actual tire characteristics, tireforce may decrease in the slip area as shown in FIG. 12. When thesolution for the four-wheel independent steering is realized in thefront-wheel and rear-wheel steering vehicle, the vehicle accelerationmay decrease by a corresponding amount.

[0119]FIG. 13 is a diagram showing solutions (vectors of forcesgenerated by the respective wheels and the vehicle body, and thesteering angles thereof) for the four-wheel-drive vehicle and thefront-wheel and rear-wheel steering vehicle when the brake is applied tothe vehicles during turning on a flat road surface having a μ of 0.85.There is no difference between the four-wheel independent steering andthe front-wheel and rear-wheel steering on the road surface having auniform μ, and both the four-wheel independent steering and thefront-wheel and rear-wheel steering obtain the solution of q_(i)=0.

[0120] Considering that almost no difference is found between thefour-wheel independent steering and the front-wheel and rear-wheelsteering vehicle on the road surface having a uniform μ, and that thedirection q_(i) of the force generated by each wheel is relativelysmall, the following control rule for the front-wheel and rear-wheelsteering may be used in critical and pre-critical traveling states.

[0121] First, in the pre-critical traveling state, the maximum value ofthe road surface μ of each wheel is used, and control when the roadsurface μ is maximum and uniform is carried out. When the wheels do nothave a uniform μ, the actual effective road friction does not becomeuniform, but the magnitude of the tire force of each wheel becomesproportional to the load distribution ratio, and the direction q_(i) ofthe force generated by each wheel becomes relatively small. Therefore,well-balanced cooperation of steering and braking, and of steering anddriving can be expected.

[0122] The control rule for the four-wheel independent steering is usedwhen the wheel having a low μ has reached its limit, namely, in the caseof the following formula (69).

F_(iReal)<γF_(i)  (69)

[0123] γ is calculated from the solution q_(i) of the recurrence formula(64) and calculated as follows. $\begin{matrix}{\gamma = {\frac{F}{{F_{1}\cos \quad q_{1}} + {F_{2}\cos \quad q_{2}} + {F_{3}\cos \quad q_{3}} + {F_{4\quad}\cos \quad q_{4}}}.}} & (70)\end{matrix}$

[0124] F_(i) is the magnitude of the tire force when μ of the roadsurface is assumed to be large, and F_(iReal) is an actual magnitude ofthe tire force of a wheel having a small μ. γF_(i) in the process ofderiving the control rule for the four-wheel independent steering can beobtained as F_(iReal). For example, when wheels 1 and 3 (the left frontwheel and the left rear wheel) have reached their limits, theconstraints for the wheels are represented as follows.

F _(1Rea1) sin q ₁ +γF ₂ sin q ₂ +F _(3Rea1) sin q ₃ +γF ₄ sin q₄=0  (71) $\begin{matrix}{{{{- a_{1}}F_{1\quad {Real}}\cos \quad q_{1}} - {a_{2}\gamma \quad F_{2}\cos \quad q_{2}} - {a_{3}F_{3\quad {Real}}\cos \quad q_{3}} - {a_{4}\gamma \quad F_{4}\cos \quad q_{4}} + {b_{1}F_{1\quad {Real}}\sin \quad q_{1}} + {b_{2}\gamma \quad F_{2}\sin \quad q_{2}} + {b_{3}F_{3\quad {Real}}\sin \quad q_{3}} + {b_{4}\gamma \quad F_{4}\sin \quad q_{4}}} = M_{z}} & (72)\end{matrix}$

 F _(1Real) sin q ₁ +γF ₂ sin q ₂ +F _(3 Real) sin q ₃ +γF ₄ sin q ₄=F  (73)

[0125] Thus, this becomes a problem of determining q_(i) which satisfiesthe formulae (71) to (73) and minimizes γ. While the nonlinearoptimization may be used, the following recurrence formula may also beused: $\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{diag}\left\lbrack {\begin{matrix}\frac{1}{\sqrt{F_{1\quad {Real}}}} & \frac{1}{\sqrt{\gamma_{0}F_{2}}} & \frac{1}{\sqrt{F_{3\quad {Real}}}} & \left. \frac{1}{\sqrt{\gamma_{0}F_{4}}} \right\rbrack\end{matrix} \cdot \begin{matrix}\left\lbrack {\sqrt{F_{1\quad {Real}}}\cos \quad q_{10}} \right. & {\sqrt{\gamma_{0}F_{2}}\cos \quad q_{20}} \\{\sqrt{F_{1\quad {Real}}}\left( {{a_{1}\sin \quad q_{10}} + {b_{1}\cos \quad q_{10}}} \right)} & {\sqrt{\gamma_{0}F_{2}}\left( {{a_{2}\sin \quad q_{20}} + {b_{2}\cos \quad q_{20}}} \right)} \\\begin{matrix}{\sqrt{F_{3\quad {Real}}}\cos \quad q_{30}} \\{\sqrt{F_{3\quad {Real}}}\left( {{a_{3}\sin \quad q_{30}} + {b_{3}\cos \quad q_{30}}} \right)}\end{matrix} & {\left. \begin{matrix}{\sqrt{\gamma_{0}F_{4}}\cos \quad q_{40}} \\{\sqrt{\gamma_{0}F_{4}}\left( {{a_{4}\sin \quad q_{40}} + {b_{4}\cos \quad q_{40}}} \right)}\end{matrix} \right\rbrack^{+} \cdot \begin{bmatrix}d_{3} \\d_{4}\end{bmatrix}}\end{matrix}} \right.}} & (74)\end{matrix}$

[0126] provided that $\begin{matrix}{{d_{3} = {{\sum\limits_{{i = 1},3}^{\quad}\quad {F_{i\quad {Real}}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{i0}}} \right)}} + {\sum\limits_{{i = 2},4}^{\quad}\quad {\gamma_{0}{F_{i\quad}\left( {{q_{i0}\cos \quad q_{i0}} - {\sin \quad q_{i0}}} \right)}}}}}{d_{4} = {M_{z} + {\sum\limits_{{i = 1},3}^{\quad}\quad {F_{i\quad {Real}}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{i0}}} \right\}}} + {\sum\limits_{{i = 2},4}^{\quad}\quad {\gamma_{0}F_{i}\left\{ {{\left( {a_{i} + {b_{i}q_{i0}}} \right)\cos \quad q_{i0}} + {\left( {{a_{i}q_{i0}} - b_{i}} \right)\sin \quad q_{i0}}} \right\}}}}}{and}{\gamma = {\frac{F - {\sum\limits_{{i = 1},3}^{\quad}\quad {F_{i\quad {Real}}\cos \quad q_{i}}}}{\sum\limits_{{i = 2},4}^{\quad}{F_{i}\cos \quad q_{i}}}.}}} & (75)\end{matrix}$

[0127] The subscript “0” in γ₀ and F_(i0) indicates a value calculatedin the previous step. Based on q_(i) thus derived, the braking force,the driving force and the steering angle can be derived from theformulae (45) to (52) for the wheel of large μ, and, for the wheel ofsmall μ, the braking force, the driving force and the steering angle canbe derived from the formulae (65) to (68), considering that the slip ofthe wheel of large μ is as follows. $\begin{matrix}{k_{i} = {\frac{3F_{i}}{K_{s}}\left( {1 - \left( {1 - \gamma} \right)^{\frac{1}{3}}} \right)}} & (76)\end{matrix}$

[0128] As described above, when the control rule for the vehicle motionaccording to the present embodiment is derived with the direction of thetire force of each wheel being used as the control variable, a nonlineartire model does not need to be included in the optimal calculation, andthe number of control variables is smaller, as compared to aconventional method in which the slip angle and the slip ratio of eachwheel are used as the control variables. Therefore, the amount ofcalculation of the control rule is smaller than that of the conventionalmethod, and the use of the control rule can be expanded to thefour-wheel independent steering, which is a system of a higher degree offreedom.

[0129] In the four-wheel-drive vehicle, the case has been described inwhich the control rule which equalizes and minimizes the effective roadfriction of each wheel is derived. This means maximizing frictionallowance of each wheel in combinations of control of each wheel forobtaining desired force and moment. Therefore, improvement in safety andresistance to failure can be expected.

[0130] Moreover, the case has been described in which the control rulewhich equalizes the effective road friction of each wheel and thecontrol rule which makes the tire force of each wheel proportional tothe load distribution are derived in the front-wheel and rear-wheelsteering vehicle in which the right and left wheels have the samesteering angle. These control rules correspond to each other when thefront-wheel and rear-wheel steering vehicle travels on the road surfaceof uniform μ, and substantially correspond to the solution of theabove-described control rule for the four-wheel-drive steering. Thus,this means that, when the front-wheel and rear-wheel steering vehicletravels on the road surface of uniform μ, these control rules maximizefriction allowance of each wheel in combinations of control of eachwheel for obtaining desired force and moment. Therefore, improvement insafety and resistance to failure can be expected.

[0131] Further, in the control rule for making the tire forceproportional to the load distribution in the front-wheel and rear-wheelsteering vehicle in which the right and left wheels have the samesteering angle, when the wheel of small μ has reached its limit offriction, the direction of the force generated by each wheel, whichdirection is derived based on the control rule for the four-wheel-drivevehicle, is realized by the front-wheel and rear-wheel steering vehicle.

[0132] Next, a second specific structure of the present embodiment usingthe above principle will be described with reference to FIG. 14.

[0133] In the present embodiment, the present invention is applied to avehicle having an electric power steering device mounted therein. Asshown in the drawing, the present embodiment is formed by a group ofsensors 10 mounted in the vehicle and including a steering angle sensorfor detecting a steering angle from a rotation angle of a steering shaftof the electric power steering device, an assist torque sensor fordetecting power assist torque from electric current passing through theelectric power steering device, a torque sensor for detecting steeringtorque, a vehicle speed sensor for detecting the speed of the vehicle,an accelerator stroke sensor for detecting an accelerator stroke, abraking effort sensor for detecting braking effort, and a wheel speedsensor for detecting the rotation speed of a wheel; a controller 12formed by microcomputers; and a cooperative braking and driving device14 which is connected to the controller 12 and cooperatively controlsthe steering angle, and braking and driving.

[0134] The controller 12 formed by the microcomputers is controlled soas to perform a plurality of functions in accordance with programsstored in advance. When the controller 12 is shown in a block functionaldiagram, the controller 12 is formed by a SAT estimating unit 16 forestimating the SAT; a target resultant force calculating unit 18 forcalculating the magnitude and direction of the target resultant force; acritical friction circle estimating unit 20 for estimating the magnitude(radius) of the critical friction circle of each wheel; a criticalresultant force estimating unit 22 for estimating the magnitude ofcritical resultant force; an effective road friction setting unit 24; amagnitude of tire force setting unit 26 for setting the magnitude oftire force; a direction of tire force setting unit 28 for setting thedirection of the tire force of each wheel; and a control unit 30connected to the cooperative braking and driving device 14.

[0135] The SAT estimating unit 16 estimates SAT based on steering torquedetected by the torque sensor and assist torque detected by the assisttorque sensor.

[0136] The target resultant force calculating unit 18 calculates themagnitude and direction of the target resultant force and the yaw momentM_(z) to be applied to the vehicle body in order to obtain a vehiclebody motion that the driver desires, from the steering angle, thevehicle speed, the accelerator stroke, the braking effort, and the likewhich have been detected by the respective sensors.

[0137] For example, a resultant force and a yaw moment that arenecessary to approximate to zero deviations from a yaw angle speed,which is a target vehicle motion state variable, and from a measuredvalue or an estimate of the slip angle of the vehicle body, can be usedas the magnitude and direction of the target resultant force and the yawmoment M_(z).

[0138] The critical friction circle estimating unit 20 estimates themagnitude of the critical friction circle of each wheel based on the SATestimated by the SAT estimating unit 16 or the wheel speed detected bythe wheel speed sensor.

[0139] The critical resultant force estimating unit 22 estimates themagnitude of the critical resultant force from the magnitude of thecritical friction circle of each wheel estimated by the criticalfriction circle estimating unit 20. The estimation may be carried out bysimply regarding the sum of the critical friction forces of therespective wheels as the critical resultant force, or by regarding asthe critical resultant force a value obtained by multiplying the sum ofthe critical friction forces of the respective wheels by a constant.Alternatively, the estimation may be carried out by first determiningthe angle q_(i) based on the formulae (21) to (24) from a direction θ ofthe target resultant force and a target yaw moment, and then calculatinga critical resultant force J based on the formula (11).

[0140] The effective road friction setting unit 24 sets as the effectiveroad friction γ the ratio of the magnitude of the target resultant forceto that of the critical resultant force. However, when the targetresultant force exceeds the critical resultant force, the effective roadfriction is set as 1.

[0141] The magnitude of tire force setting unit 26 sets the magnitude ofthe tire force used at each wheel by multiplying the critical frictioncircle by the effective road friction.

[0142] The direction of the tire force setting unit 28 uses thedirection θ of the target resultant force and the magnitude γF_(i) ofthe tire force of each wheel used, to determine the angle q_(i) based onthe following formula obtained by replacing F_(i) in the formulae (20)to (24) by γF_(i), and outputs the magnitude γF_(i) and the direction(q_(i)+0) of the tire force of each wheel: $\begin{matrix}\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}q_{1} \\q_{2}\end{matrix} \\q_{3}\end{matrix} \\q_{4}\end{bmatrix} = {{{diag}\begin{bmatrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{bmatrix}} \cdot}} \\{{\begin{bmatrix}\sqrt{F_{1}} & \sqrt{F_{2}} & \sqrt{F_{3}} & \sqrt{F_{4}} \\{b_{1}\sqrt{F_{1}}} & {b_{2}\sqrt{F_{2}}} & {b_{3}\sqrt{F_{3}}} & {b_{4}\sqrt{F_{4}}}\end{bmatrix}^{+} \cdot}} \\{\left\lbrack {\frac{M_{z}}{\gamma} + \left( {{a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \right)} \right\rbrack} \\{= {\frac{\frac{M_{z}}{\gamma} + \left( {{a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \right)}{{b_{1}c_{1}F_{1}} + {b_{2}c_{2}F_{2}} + {b_{3}c_{3}F_{3}} + {b_{4}c_{4}F_{4}}} \cdot \begin{bmatrix}\begin{matrix}\begin{matrix}c_{1} \\c_{2}\end{matrix} \\c_{3}\end{matrix} \\c_{4}\end{bmatrix}}}\end{matrix} & (77)\end{matrix}$

[0143] provided that

c₁=(b ₁ −b ₂)F ₂+(b ₁ −b ₃)F ₃+(b ₁ −b ₄)F ₄  (78)

c ₂=(b ₂ −b ₁)F ₁+(b ₂ −b ₃)F ₃+(b ₂ −b ₄)F ₄  (79)

c ₃=(b ₃ −b ₁)F ₁+(b ₃ −b ₂)F ₂+(b ₃ −b ₄)F ₄  (80)

c ₄=(b ₄ −b ₁)F ₁+(b ₄ −b ₂)F ₂+(b ₄ −b ₃)F ₃  (81)

[0144] The control unit 30 determines the steering angle and the brakingforce, or the steering angle and the driving force of each wheel basedon the magnitude γF_(i) and the direction (q_(i)+θ) of the tire force ofeach wheel, and controls the steering device and the braking actuator,or the steering device and the driving actuator. The braking force orthe driving force of each wheel can be derived as follows from themagnitude γF_(i) and the direction (q_(i)+θ) of the tire force of eachwheel.

F _(xi) =γF _(i) cos (q+θ)  (82)

[0145] Similarly, lateral force of each wheel can be derived from thefollowing formula.

F _(γi) =γF _(i) sin(q _(i)+θ)  (83)

[0146] The steering angle of each wheel can be calculated, for example,based on the brush model and the vehicle motion model. The brush modelis the model which describes the characteristic of the tire force basedon the theoretical formula. Assuming that the tire force is generated inaccordance with the brush model, the slip angle β_(i) can be calculatedby the following formula, using the critical friction circle F_(i), theeffective road friction γ, and the direction of the tire force(q_(i)+θ): $\begin{matrix}{\beta_{i} = {\tan^{- 1}\left\lbrack {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{i}}{\sin \left( {q_{i} + \theta} \right)}}{1 - {k_{i}{\cos \left( {q_{i} + \theta} \right)}}}} \right\rbrack}} & (84)\end{matrix}$

[0147] provided that $\begin{matrix}{k_{i} = {{\frac{3F_{i}}{K_{s}}\left\lbrack {1 - \left( {1 - \gamma} \right)^{1/3}} \right\rbrack}.}} & (85)\end{matrix}$

[0148] In the above formulae, K_(s) represents driving stiffness, andK_(β) represents cornering stiffness. Further, the steering angle ofeach wheel is calculated from the slip angle β_(i) based on the vehiclemotion model. Namely, the steering angle can be calculated as follows byusing the yaw angle speed r₀ and the vehicle body slip angle β₀, whichare calculated as the target vehicle motion state variables from thevehicle speed v, the steering angle, the accelerator stroke, the brakingeffort, and the like. $\begin{matrix}{\delta_{1} = {\beta_{0} + {\frac{L_{f}}{v}r_{0}} - \beta_{1}}} & (86) \\{\delta_{2} = {\beta_{0} + {\frac{L_{f}}{v}r_{0}} - \beta_{2}}} & (87) \\{\delta_{3} = {\beta_{0} - {\frac{L_{r}}{v}r_{0}} - \beta_{3}}} & (88) \\{\delta_{4} = {\beta_{0} - {\frac{L_{r}}{v}r_{0}} - \beta_{4}}} & (89)\end{matrix}$

[0149] When the cooperative control of steering and braking and ofsteering and driving is carried out based on this type of control, theeffective road friction of each wheel can be made uniform all the time,and motion performance allowing the greatest robust against disturbancesuch as a road surface or cross wind can be exhibited.

[0150] Further, in the present embodiment, the effective road frictioncan be independently set for the front wheel and the rear wheel. In thiscase, the following formula is constructed, wherein the effective roadfrictions of the front wheel and the rear wheel are γ_(f) and γ_(r)respectively: $\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}q_{1} \\q_{2}\end{matrix} \\q_{3}\end{matrix} \\q_{4}\end{bmatrix} = {\frac{M_{z} + {a_{1}\gamma_{f}F_{1}} + {a_{2}\gamma_{f}F_{2}} + {a_{3}\gamma_{r}F_{3}} + {a_{4}\gamma_{r}F_{4}}}{{b_{1}c_{1}\gamma_{f}F_{1}} + {b_{2}c_{2}\gamma_{f}F_{2}} + {b_{3}c_{3}\gamma_{r}F_{3}} + {b_{4}c_{4}\gamma_{r}F_{4}}} \cdot \quad \begin{bmatrix}\begin{matrix}\begin{matrix}c_{1} \\c_{2}\end{matrix} \\c_{3}\end{matrix} \\c_{4}\end{bmatrix}}} & (100)\end{matrix}$

[0151] provided that

c ₁=(b ₁ −b ₂)γ_(f) F ₂+(b ₁ −b ₃)γ_(r) F ₃+(b ₁ −b ₄)γ_(r) F ₄  (101)

c ₂=(b ₂ −b ₁)γ_(f) F ₁+(b ₂ −b ₃)γ_(r) F ₃+(b ₂ −b ₄)γ_(r) F ₄  (102)

c ₃=(b ₃ −b ₁)γ_(f) F ₁+(b ₃ −b ₂)γ_(f) F ₂+(b ₃ −b ₄)γ_(r) F ₄  (103)

c ₄=(b ₄ −b ₁)γ_(f) F ₁+(b ₄ −b ₂)γfF ₂+(b ₄ −b ₃)γ_(r) F ₃  (104)

[0152] As described above, the effective road friction is independentlyset for the front wheel and the rear wheel. Namely, the effective roadfrictions of the front wheel and the rear wheel are made to differ fromeach other. As a result, for example, by setting the effective roadfriction of the rear wheel smaller than that of the front wheel,friction allowance of the rear wheel is increased, thereby realizingmotion control which is highly effective in suppressing spin andemphasizes safety.

[0153] Further, the formulae (101) to (104) respectively represent thesum of products of the distance from the position of the object wheel tothe position of the other wheel in the direction of the resultant forceof the vehicle body, and the magnitude of the tire force to beoutputted, which tire force is obtained in consideration of theeffective road friction. As a result, the formulae (101) to (104)represent that the angle formed by the direction of the tire force ofeach wheel and the direction of the resultant force of the vehicle bodyis proportional to the sum of the products of the distance from theposition of the object wheel to the position of the other wheel in thedirection of the resultant force of the vehicle body, and the magnitudeof the tire force to be outputted.

[0154] Further, considering that a_(i) or b_(i) is a function of thedirection θ of the resultant force of the vehicle body, the formulae(101) to (104) represent that the angle formed by the direction of theforce generated by each wheel and the direction of the resultant forceof the vehicle body can be described as a function of the direction ofthe resultant force of the vehicle body and the magnitude of the tireforce of each wheel to be outputted.

[0155] Next, an embodiment will be described in which the presentinvention is applied to the normal four-wheel vehicle, which iscontrolled such that the steering angle is the same for the right andleft wheels.

[0156] In the case of the conventional four-wheel vehicle having thesame steering angle for the right and left wheels, constraintsrepresented by the following formulae (105) and (106) and indicatingthat the right and left wheels have the same slip angle are furtheradded. These constraints can further be organized into the followingformulae (107) and (108). $\begin{matrix}{{\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{1}}{\sin \left( {q_{1} + \theta} \right)}}{1 - {k_{1}{\cos \left( {q_{1} + \theta} \right)}}}} \right)} = {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{2}}{\sin \left( {q_{2} + \theta} \right)}}{1 - {k_{2}{\cos \left( {q_{2} + \theta} \right)}}}} \right)}} & (105) \\{{\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{3}}{\sin \left( {q_{3} + \theta} \right)}}{1 - {k_{3}{\cos \left( {q_{3} + \theta} \right)}}}} \right)} = {\tan^{- 1}\left( {\frac{K_{s}}{K_{\beta}} \cdot \frac{{- k_{4}}{\sin \left( {q_{4} + \theta} \right)}}{1 - {k_{4}{\cos \left( {q_{4} + \theta} \right)}}}} \right)}} & (106)\end{matrix}$

 k ₂ sin(q ₂+θ)−k ₁ sin (q ₁+θ)−k ₁ k ₂ sin (q ₂ −q ₁)=0  (107)

k ₄ sin (q ₄+θ)−k ₃ sin (q ₃+θ)−k ₃ k ₄ sin(q ₄ −q ₃)=0  (108)

[0157] After primary approximation, these constraints are represented bythe following formulae (109) and (110).

−k ₁(cos θ−k ₂)q ₁ +k ₂(cos θ−k ₁)q ₂=(k ₁ −k ₂)sin θ  (109)

−k ₃(cos θ−k ₄)q ₃ +k ₄(cos θ−k ₃)q ₄=(k ₃ −k ₄)sin θ  (110)

[0158] Furthermore, these constraints are represented by using p_(i) asfollows. $\begin{matrix}{{{{- \frac{k_{1}}{\sqrt{F_{1}}}}\left( {{\cos \quad \theta} - k_{2}} \right)p_{1}} + {\frac{k_{2}}{\sqrt{F_{2}}}\left( {{\cos \quad \theta} - k_{1}} \right)p_{2}}} = {\left( {k_{1} - k_{2}} \right)\sin \quad \theta}} & (111) \\{{{{- \frac{k_{3}}{\sqrt{F_{3}}}}\left( {{\cos \quad \theta} - k_{4}} \right)p_{3}} + {\frac{k_{4}}{\sqrt{F_{4}}}\left( {{\cos \quad \theta} - k_{3}} \right)p_{4}}} = {\left( {k_{3} - k_{4}} \right)\sin \quad \theta}} & (112)\end{matrix}$

[0159] By considering other constraints represented by the followingformulae (113) and (114) when the vehicle travels with an effective roadfriction γ, the angle q_(i) formed by the direction of the resultantforce and the force generated by a single wheel can be uniquelydetermined as shown in the following formula (115).

{square root}{square root over (F ₁)}p ₁+{square root}{square root over(F ₂)}p ₂+{square root}{square root over (F ₃)}p ₃+{square root}{squareroot over (F ₄)}p ₄  (113) $\begin{matrix}{{{b_{1}\sqrt{F_{1}}p_{1}} + {b_{2}\sqrt{F_{2}}p_{2}} + {b_{3}\sqrt{F_{3}}p_{3}} + {b_{4}\sqrt{F_{4}}p_{4}}} = {\frac{M_{z}}{\gamma} + \left( {{a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}} \right)}} & (114)\end{matrix}$

$\begin{matrix}\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}q_{1} \\q_{2}\end{matrix} \\q_{3}\end{matrix} \\q_{4}\end{bmatrix} = {{{diag}\begin{bmatrix}\frac{1}{\sqrt{F_{1}}} & \frac{1}{\sqrt{F_{2}}} & \frac{1}{\sqrt{F_{3}}} & \frac{1}{\sqrt{F_{4}}}\end{bmatrix}} \cdot}} \\{{~~}{\begin{bmatrix}\sqrt{F_{1}} & \sqrt{F_{2}} & \sqrt{F_{3}} & \sqrt{F_{4}} \\{b_{1}\sqrt{F_{1}}} & {b_{2}\sqrt{F_{2}}} & {b_{3}\sqrt{F_{3}}} & {b_{4}\sqrt{F_{4}}} \\{{- \frac{k_{1}}{\sqrt{F_{1}}}}\left( {{\cos \quad \theta} - k_{2}} \right)} & {\frac{k_{2}}{\sqrt{F_{2}}}\left( {{\cos \quad \theta} - k_{1}} \right)} & 0 & 0 \\0 & 0 & {{- \frac{k_{3}}{\sqrt{F_{3}}}}\left( {{\cos \quad \theta} - k_{4}} \right)} & {\frac{k_{4}}{\sqrt{F_{4}}}\left( {{\cos \quad \theta} - k_{3}} \right)}\end{bmatrix}^{- 1} \cdot}} \\{\quad \begin{bmatrix}\begin{matrix}0 \\{\frac{M_{z}}{\gamma} + {a_{1}F_{1}} + {a_{2}F_{2}} + {a_{3}F_{3}} + {a_{4}F_{4}}}\end{matrix} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta} \\{\left( {k_{3} - k_{4}} \right)\sin \quad \theta}\end{bmatrix}}\end{matrix} & (115)\end{matrix}$

[0160] The braking force, the driving force and the steering angle ofeach wheel at this time are calculated by the formulae (82) and (84) to(89). However, the same value is calculated in the formulae (87) and(89) for the steering angles of the right and left wheels. The valueq_(i) obtained by formula (115) is calculated by primary approximationand may also be used as an initial value to numerically solve acorresponding nonlinear equation, and the control is carried out basedon the solution of the equation. In this case, control of higheraccuracy can be achieved.

[0161] When the cooperation of the steering control and the braking anddriving control is carried out based on this type of control, theeffective road friction of each wheel can be made uniform all the time,and motion performance allowing the greatest robust against disturbancesuch as a road surface or cross wind can be exhibited.

[0162] Moreover, even when the same steering angle is used for the rightand left wheels, the effective road friction of each wheel can be setindependently. In this case, the angle q_(i) formed by the direction ofthe resultant force and the force generated by a single wheel isrepresented by the following formula. $\begin{matrix}\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}q_{1} \\q_{2}\end{matrix} \\q_{3}\end{matrix} \\q_{4}\end{bmatrix} = {{{diag}\begin{bmatrix}\frac{1}{\sqrt{F_{1}\gamma_{f}}} & \frac{1}{\sqrt{F_{2}\gamma_{f}}} & \frac{1}{\sqrt{F_{3}\gamma_{r}}} & \frac{1}{\sqrt{F_{4}\gamma_{r}}}\end{bmatrix}} \cdot}} \\{\quad {\begin{bmatrix}\sqrt{F_{1}\gamma_{f}} & \sqrt{F_{2}\gamma_{f}} & \sqrt{F_{3}\gamma_{r}} & \sqrt{F_{4}\gamma_{r}} \\{b_{1}\sqrt{F_{1}\gamma_{f}}} & {b_{2}\sqrt{F_{2}\gamma_{f}}} & {b_{3}\sqrt{F_{3}\gamma_{r}}} & {b_{4}\sqrt{F_{4}\gamma_{r}}} \\{{- \frac{k_{1}}{\sqrt{F_{1}\gamma_{f}}}}\left( {{\cos \quad \theta} - k_{2}} \right)} & {\frac{k_{2}}{\sqrt{F_{2}\gamma_{f}}}\left( {{\cos \quad \theta} - k_{1}} \right)} & 0 & 0 \\0 & 0 & {{- \frac{k_{3}}{\sqrt{F_{3}\gamma_{r}}}}\left( {{\cos \quad \theta} - k_{4}} \right)} & {\frac{k_{4}}{\sqrt{F_{4}\gamma_{r}}}\left( {{\cos \quad \theta} - k_{3}} \right)}\end{bmatrix}^{- 1} \cdot}} \\{\quad \begin{bmatrix}\begin{matrix}0 \\{M_{z} + \left( {{a_{1}F_{1}\gamma_{f}} + {a_{2}F_{2}\gamma_{f}} + {a_{3}F_{3}\gamma_{r}} + {a_{4}F_{4}\gamma_{r}}} \right)}\end{matrix} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta} \\{\left( {k_{3} - k_{4}} \right)\sin \quad \theta}\end{bmatrix}} \\{\quad {= {\begin{bmatrix}{F_{1}\gamma_{f}} & {F_{2}\gamma_{f}} & {F_{3}\gamma_{r}} & {F_{4}\gamma_{r}} \\{b_{1}F_{1}\gamma_{f}} & {b_{2}F_{2}\gamma_{f}} & {b_{3}F_{3}\gamma_{r}} & {b_{4}F_{4}\gamma_{r}} \\{- {k_{1}\left( {{\cos \quad \theta} - k_{2}} \right)}} & {k_{2}\left( {{\cos \quad \theta} - k_{1}} \right)} & 0 & 0 \\0 & 0 & {- {k_{3}\left( {{\cos \quad \theta} - k_{4}} \right)}} & {k_{4}\left( {{\cos \quad \theta} - k_{3}} \right)}\end{bmatrix}^{- 1} \cdot}}} \\{\quad \begin{bmatrix}\begin{matrix}0 \\{M_{z} + \left( {{a_{1}F_{1}\gamma_{f}} + {a_{2}F_{2}\gamma_{f}} + {a_{3}F_{3}\gamma_{r}} + {a_{4}F_{4}\gamma_{r}}} \right)}\end{matrix} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta} \\{\left( {k_{3} - k_{4}} \right)\sin \quad \theta}\end{bmatrix}}\end{matrix} & (116)\end{matrix}$

[0163] The present embodiment can also be applied to a vehiclestructured such that only the front wheels or the rear wheels can beindependently steered. For example, in a case of a vehicle in which eachof the rear wheels can be independently steered, the angle q_(i)formedby the direction of the resultant force and the force generated by asingle wheel can be represented by the following formula.$\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{{diag}\left\lbrack \quad \begin{matrix}\frac{1}{\sqrt{F_{1}\gamma_{f}}} & \frac{1}{\sqrt{F_{2}\gamma_{f}}} & \frac{1}{\sqrt{F_{3}\gamma_{r}}} & \frac{1}{\sqrt{F_{4}\gamma_{r}}}\end{matrix}\quad \right\rbrack} \cdot \begin{bmatrix}\sqrt{F_{1}\gamma_{f}} & \sqrt{F_{2}\gamma_{f}} & \sqrt{F_{3}\gamma_{r}} & \sqrt{F_{4}\gamma_{r}} \\{b_{1}\sqrt{F_{1}\gamma_{f}}} & {b_{2}\sqrt{F_{2}\gamma_{f}}} & {b_{3}\sqrt{F_{3}\gamma_{r}}} & {b_{4}\sqrt{F_{4}\gamma_{r}}} \\{{- \frac{k_{1}}{\sqrt{F_{1}\gamma_{f}}}}\left( {{\cos \quad \theta} - k_{2}} \right)} & {\frac{k_{2}}{\sqrt{F_{2}\gamma_{f}}}\left( {{\cos \quad \theta} - k_{1}} \right)} & 0 & 0\end{bmatrix}^{+} \cdot {\quad\left\lbrack \left. \quad\begin{matrix}{M_{z} + \left( {{a_{1}F_{1}\gamma_{f}} + {a_{2}F_{2}\gamma_{f}} + {a_{3}F_{3}\gamma_{r}} + {a_{4}F_{4}\gamma_{r}}} \right)} \\{\left( {k_{1} - k_{2}} \right)\sin \quad \theta}\end{matrix} \right\rbrack \right.}}} & (117)\end{matrix}$

[0164] The magnitude of the tire force in the present embodiment canalso be represented by the magnitude of the friction circle.

What is claimed is:
 1. A vehicle control method comprising: calculatinga physical quantity which relates to a tire force of each wheel andoptimizes an effective road friction of each wheel, based on a targetresultant force to be applied to a vehicle body in order to obtainvehicle body motion that a driver desires, and a constraint including asparameters a magnitude of a critical friction circle of each wheel;calculating, based on the calculated physical quantity relating to thetire force of each wheel, a first control variable for controlling atleast one of braking force and driving force of each wheel, or a secondcontrol variable for controlling the first control variable and asteering angle of each wheel; and controlling (A) the at least one ofbraking force and driving force of each wheel based on the first controlvariable, or controlling (A) the at least one of braking force anddriving force of each wheel and (B) the steering angle of each wheelbased on the first and second control variables.
 2. The method of claim1, wherein the constraint is represented by a formula indicating that noresultant force is generated in a direction orthogonal to a direction ofthe target resultant force, and a formula indicating that a momentaround the center of gravity of the vehicle is equal to a desiredmoment.
 3. The method of claim 2, wherein the constraint is representedby formulae, the number of which is less than that of wheels, or alinearized formula.
 4. The method of claim 2, wherein: the targetresultant force is represented by a secondary performance functionincluding the magnitude of the critical friction circle of each wheeland the physical quantity relating to the tire force of each wheel; anda physical quantity relating to the tire force of each wheel, whichphysical quantity satisfies a first approximation formula of a formuladefining the constraint and optimizes the secondary performancefunction, is calculated as the physical quantity which relates to thetire force of each wheel and optimizes the effective road friction ofeach wheel.
 5. The method of claim 2, wherein: the target resultantforce is represented by a secondary performance function including themagnitude of the critical friction circle of each wheel and the physicalquantity relating to the tire force of each wheel; and a physicalquantity relating to the tire force of each wheel, which physicalquantity satisfies a first approximation formula of a formula definingthe constraint and optimizes the secondary performance function, iscalculated as an initial value; the formula defining the constraint islinearized by using the calculated initial value; a physical quantityrelating to the tire force of each wheel, which physical quantitysatisfies the linearized formula of the constraint and optimizes thesecondary performance function, is calculated as an approximatesolution; and the physical quantity which relates to the tire force ofeach wheel and optimizes the effective road friction of each wheel iscalculated by using the calculated approximate solution as the initialvalue to repeat the linearization of the formula defining the constraintand the calculation of the approximate solution.
 6. The method of claim5, wherein the formula defining the constraint is linearized by Taylorexpansion around the initial value or the approximate solution.
 7. Themethod of claim 1, wherein: the physical quantity relating to the tireforce is a direction of the tire force; and the effective road frictionof each wheel, the calculated direction of the tire force of each wheel,and the magnitude of the critical friction circle of each wheel are usedto calculate a slip angle based on a brush model, and the calculatedslip angle is used to calculate the second control variable based on avehicle motion model.
 8. The method of claim 1, wherein the magnitude ofthe critical friction circle of each wheel is determined based on anestimate or a virtual value of μ of each wheel and a load of each wheel.9. The method of claim 1, wherein: the physical quantity relating to thetire force is a direction of the tire force; and the direction of thetire force which optimizes the effective road friction of each wheel isone of a direction of the tire force which uniformly minimizes theeffective road friction of each wheel, a direction of the tire forcewhich makes the effective road friction of a front wheel differ fromthat of a rear wheel, and a direction of the tire force which makes amagnitude of the tire force of each wheel proportional to the load ofthe wheel.
 10. The method of claim 9, further comprising using, for awheel having a small μ, the magnitude of the critical friction circle asthe magnitude of the tire force, and using, for a wheel having a largeμ, the magnitude of the tire force which minimizes the effective roadfriction, when the magnitude of the tire force proportional to the loadof the wheel cannot be obtained because each wheel has a different μwith respect to a road surface.
 11. The method of claim 1, wherein thesteering angle is controlled so as to be the same for the right and leftwheels.
 12. The method of claim 1, wherein the effective road frictionis represented by a magnitude of the target resultant force relative toa magnitude of a critical resultant force obtained from the magnitude ofthe critical friction circle of each wheel.
 13. The method of claim 7,wherein the direction of the tire force which is generated by each wheelis defined as a value, that is the sum of products, which are calculatedfor all other wheels, of a distance from the position of an object wheelto the position of the other wheel in the direction of the resultantforce, and the magnitude of the critical friction circle of the otherwheel, with the direction of the resultant force acting on the vehiclebody as the resultant force of the tire forces of the respective wheelsbeing used as a reference.
 14. A vehicle control apparatus comprising:target resultant force calculating means for calculating a targetresultant force to be applied to a vehicle body in order to obtain avehicle body motion that a driver desires; critical friction circleestimating means for estimating a magnitude of a critical frictioncircle of each wheel; tire force calculating means for calculating aphysical quantity which relates to a tire force of each wheel andoptimizes an effective road friction of each wheel, based on the targetresultant force and a constraint including as parameters the magnitudeof the critical friction circle of each wheel; control variablecalculating means for calculating, based on the calculated physicalquantity relating to the tire force of each wheel, a first controlvariable for controlling at least one of braking force and driving forceof each wheel, or a second control variable for controlling the firstcontrol variable and a steering angle of each wheel; and control meansfor controlling (A) the at least one of braking force and driving forceof each wheel based on the first control variable, or controlling (A)the at least one of braking force and driving force of each wheel and(B) the steering angle of each wheel based on the first and secondcontrol variables.
 15. The apparatus of claim 14, wherein the constraintis represented by a formula indicating that no resultant force isgenerated in a direction orthogonal to a direction of the targetresultant force, and a formula indicating that a moment around thecenter of gravity of the vehicle is equal to a desired moment.
 16. Theapparatus of claim 15, wherein the constraint is represented byformulae, the number of which is less than that of wheels, or alinearized formula.
 17. The apparatus of claim 15, wherein: the targetresultant force is represented by a secondary performance functionincluding the magnitude of the critical friction circle of each wheeland the physical quantity relating to the tire force of each wheel; andthe tire force calculating means calculates a physical quantity relatingto the tire force of each wheel, which physical quantity satisfies afirst approximation formula of a formula defining the constraint andoptimizes the secondary performance function, as the physical quantitywhich relates to the tire force of each wheel and optimizes theeffective road friction of each wheel.
 18. The apparatus of claim 15,wherein: the target resultant force is represented by a secondaryperformance function including the magnitude of the critical frictioncircle of each wheel and the physical quantity relating to the tireforce of each wheel; and the tire force calculating means calculates asan initial value a physical quantity relating to the tire force of eachwheel, which physical quantity satisfies a first approximation formulaof a formula defining the constraint and optimizes the secondaryperformance function, linearizes the formula defining the constraint byusing the calculated initial value, calculates as an approximatesolution a physical quantity relating to the tire force of each wheel,which physical quantity satisfies the linearized formula of theconstraint and optimizes the secondary performance function, andcalculates the physical quantity which relates to the tire force of eachwheel and optimizes the effective road friction of each wheel by usingthe calculated approximate solution as the initial value to repeat thelinearization of the formula defining the constraint and the calculationof the approximate solution.
 19. The apparatus of claim 18, wherein thetire force calculating means linearizes the formula defining theconstraint by Taylor expansion around the initial value or theapproximate solution.
 20. The apparatus of claim 14, wherein: thephysical quantity relating to the tire force is a direction of the tireforce; and the control variable calculating means calculates a slipangle based on a brush model by using the effective road friction ofeach wheel, the calculated direction of the tire force of each wheel,and the magnitude of the critical friction circle of each wheel, andcalculates the second control variable based on a vehicle motion modelby using the calculated slip angle.
 21. The apparatus of claim 14,wherein the critical friction circle estimating means determines themagnitude of the critical friction circle of each wheel based on anestimate or a virtual value of μ of each wheel and a load of each wheel.22. The apparatus of claim 14, wherein: the physical quantity relatingto the tire force is a direction of the tire force; and the direction ofthe tire force which optimizes the effective road friction of each wheelis one of a direction of the tire force which uniformly minimizes theeffective road friction of each wheel, a direction of the tire forcewhich makes the effective road friction of a front wheel differ fromthat of a rear wheel, and a direction of the tire force which makes themagnitude of the tire force of each wheel proportional to a load of thewheel.
 23. The apparatus of claim 22, wherein, when the magnitude of thetire force proportional to the load of the wheel cannot be obtainedbecause each wheel has a different μ with respect to a road surface, themagnitude of the critical friction circle is used as the magnitude ofthe tire force for a wheel having a small μ, and the magnitude of thetire force which minimizes the effective road friction is used for awheel having a large μ.
 24. The apparatus of claim 14, wherein thecontrol means controls the steering angle so that the steering angle isthe same for the right and left wheels.
 25. The apparatus of claim 14,wherein the effective road friction is represented by a magnitude of thetarget resultant force relative to a magnitude of a critical resultantforce obtained from the magnitude of the critical friction circle ofeach wheel.
 26. The apparatus of claim 20, wherein the direction of thetire force which is generated by each wheel is defined as a value, thatis a sum of products, which are calculated for all other wheels, of adistance from the position of an object wheel to the position of theother wheel in the direction of the resultant force, and the magnitudeof the critical friction circle of the other wheel, with the directionof the resultant force acting on the vehicle body as the resultant forceof the tire forces of the respective wheels being used as a reference.27. A vehicle control apparatus comprising: target resultant forcecalculating means for calculating a target resultant force to be appliedto a vehicle body in order to obtain a vehicle body motion that a driverdesires; critical friction circle estimating means for estimating amagnitude of a critical friction circle of each wheel; criticalresultant force estimating means for estimating a critical resultantforce based on the magnitude of the critical friction circle of eachwheel estimated by the critical friction circle estimating means;effective road friction setting means for setting a ratio of the targetresultant force to the critical resultant force as an effective roadfriction; magnitude of tire force setting means for setting a magnitudeof a tire force used at each wheel, which tire force is obtained bymultiplying the magnitude of the critical friction circle of each wheelby the effective road friction; direction of tire force setting meansfor setting a direction of the tire force generated by each wheel basedon a value, that is a sum of products, which are calculated for allother wheels, of a distance from the position of an object wheel to theposition of the other wheel in a direction of the resultant force, andthe magnitude of the critical friction circle of the other wheel, withthe direction of the resultant force acting on the vehicle body as theresultant force generated by the tire force of each wheel being used asa reference; and control means for controlling a steering angle of eachwheel and at least one of braking force and driving force of each wheelbased on the magnitude and direction of the tire force which have beenset.
 28. The apparatus of claim 27, wherein the control means comprises:means for calculating, based on the calculated direction and magnitudeof the tire force of each wheel, a first control variable forcontrolling at least one of braking force and driving force of eachwheel, or a second control variable for controlling the first controlvariable and the steering angle of each wheel; and means for controlling(A) the at least one of braking force and driving force of each wheelbased on the first control variable, or controlling (A) the at least oneof braking force and driving force of each wheel and (B) the steeringangle of each wheel based on the first and second control variables.